[Math.AG] 13 Nov 2006 N Oti H Otgnrlfruain N Opeeproofs

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Contents Introduction i 1. Patchworking plane real algebraic curves 2 1.1. The case of smallest patches 2 1.2. Logarithmic asymptotes of a curve 4 1.3. Charts of polynomials 7 1.4. Recovering the topology of a curve from a chart of the polynomial 10 1.5. Patchworking charts 13 1.6. Patchworking polynomials 13 1.7. The Main Patchwork Theorem 14 1.8. ConstructionofM-curvesofdegree6 14 1.9. Behavior of curve VRR2 (bt) as t → 0 15 1.10. Patchworking as smoothing of singularities 17 1.11. Evolvings of singularities 18 2. Toric varieties and their hypersurfaces 21 2.1. Algebraic tori KRn 21 2.2. Polyhedra and cones 22 2.3. Affine toric variety 23 2.4. Quasi-projective toric variety 25 2.5. Hypersurfaces of toric varieties 27 3. Charts 29 3.1. Space R+∆ 29 3.2. Charts of K∆ 29 3.3. Charts of L-polynomials 30 4. Patchworking 32 4.1. Patchworking L-polynomials 32 4.2. Patchworking charts 32 4.3. The Main Patchwork Theorem 32 5. Perturbations smoothing a singularity of hypersurface 35 5.1. Singularities of hypersurfaces 35 5.2. Evolving of a singularity 36 5.3. Charts of evolving 38 5.4. Construction of evolvings by patchworking 38 6. Approximation of hypersurfaces of KRn 39 6.1. Sufficient truncations 39 6.2. Domains of ε-sufficiencyofface-truncation 40 6.3. The main Theorem on logarithmic asymptotes of hypersurface 42 6.4. Proof of Theorem 6.3.A 42 6.5. Modification of Theorem 6.3.A 43 6.6. Charts of L-polynomials 44 6.7. Structure of VKRn (bt) with small t 44 6.8. Proof of Theorem 6.7.A 45 6.9. An alternative proof of Theorem 4.3.A 46 References 47 2 OLEG VIRO

1. Patchworking plane real algebraic curves This Section is introductory. I explain the character of results staying in the framework of plane curves. A real exposition begins in Section 2. It does not depend on Section 1. To a reader who is motivated enough and does not like informal texts without proofs, I would recommend to skip this Section.

1.1. The case of smallest patches. We start with the special case of the patchworking. In this case the patches are so simple that they do not demand a special care. It puriﬁes the construction and makes it a straight bridge between combinatorial geometry and real algebraic geometry. 1.1.A (Initial Data). Let m be a positive integer number [it is the degree of the curve under construction]. Let ∆ be the triangle in R2 with vertices (0, 0), (m, 0), (0,m)[it is a would-be Newton diagram of the equation]. Let T be a triangulation of ∆ whose vertices have integer coordinates. Let the vertices of T be equipped with signs; the sign (plus or minus) at the vertex with coordinates (i, j) is denoted by σi,j . See Figure 1.

+ - +

Figure 1.

2 2 For ε,δ = ±1 denote the reﬂection R → R : (x, y) 7→ (εx,δy) by Sε,δ. For a 2 set A ⊂ R , denote Sε,δ(A) by Aε,δ (see Figure 2). Denote a quadrant {(x, y) ∈ 2 R | εx > 0,δy > 0} by Qε,δ.

Figure 2. PATCHWORKING REAL ALGEBRAIC VARIETIES 3

The following construction associates with Initial Data 1.1.A above a piecewise linear curve in the projective plane.

1.1.B (Combinatorial patchworking). Take the square ∆∗ made of ∆ and its mirror images ∆+−, ∆−+ and ∆−−. Extend the triangulation T of ∆ to a triangulation T∗ of ∆∗ symmetric with respect to the coordinate axes. Extend the distribution of signs σi,j to a distribution of signs on the vertices of the extended triangulation i j which satisfies the following condition: σi,j σεi,δj ε δ = 1 for any vertex (i, j) of T and ε,δ = ±1. (In other words, passing from a vertex to its mirror image with respect to an axis we preserve its sign if the distance from the vertex to the axis is even, and change the sign if the distance is odd.)2 If a triangle of the triangulation T∗ has vertices of different signs, draw the midline separating the vertices of different signs. Denote by L the union of these midlines. It is a collection of polygonal lines contained in ∆∗. Glue by S−− the opposite sides of ∆∗. The resulting space ∆¯ is homeomorphic to the projective plane RP 2. Denote by L¯ the image of L in ∆¯ .

+ + + - +

+ +

Figure 3. Combinatorial patchworking of the initial data shown in Figure 1

Let us introduce a supplementary assumption: the triangulation T of ∆ is convex. It means that there exists a convex piecewise linear function ν : ∆ → R which is linear on each triangle of T and not linear on the union of any two triangles of T. A function ν with this property is said to convexify T. In fact, to stay in the frameworks of algebraic geometry we need to accept an additional assumption: a function ν convexifying T should take integer value on each vertex of T. Such a function is said to convexify T over Z. However this

2More sophisticated description: the new distribution should satisfy the modular property: ∗ i j i j g (σi,j x y ) = σg(i,j)x y for g = Sεδ (in other words, the sign at a vertex is the sign of the corresponding monomial in the quadrant containing the vertex). 4 OLEG VIRO additional restriction is easy to satisfy. A function ν : ∆ → R convexifying T is characterized by its values on vertices of T. It is easy to see that this provides a natural identiﬁcation of the set of functions convexifying T with an open convex cone of RN where N is the number of vertices of T. Therefore if this set is not empty, then it contains a point with rational coordinates, and hence a point with integer coordinates.

1.1.C (Polynomial patchworking). Given Initial Data m, ∆, T and σi,j as above and a function ν convexifying T over Z. Take the polynomial i j ν(i,j) b(x,y,t)= σi,j x y t . (i, j) Xruns over vertices of T and consider it as a one-parameter family of polynomials: set bt(x, y) = b(x,y,t). Denote by Bt the corresponding homogeneous polynomials: m Bt(x0, x1, x2)= x0 bt(x1/x0, x2/x0).

1.1.D (Patchwork Theorem). Let m, ∆, T and σi,j be an initial data as above and ν a function convexifying T over Z. Denote by bt and Bt the non-homogeneous and homogeneous polynomials obtained by the polynomial patchworking of these initial data and by L and L¯ the piecewise linear curves in the square ∆∗ and its quotient space ∆¯ respectively obtained from the same initial data by the combinatorial patchworking. Then there exists t0 > 0 such that for any t ∈ (0,t0] the equation bt(x, y)=0 2 2 deﬁnes in the plane R a curve ct such that the pair (R ,ct) is homeomorphic to the pair (∆∗,L) and the equation Bt(x0, x1, x2)=0 deﬁnes in the real projective 2 plane a curve Ct such that the pair (RP , Ct) is homeomorphic to the pair (∆¯ , L¯). Example 1.1.E. Construction of a curve of degree 2 is shown in Figure 3. The broken line corresponds to an ellipse. More complicated examples with a curves of degree 6 are shown in Figures 4, 5. For more general version of the patchworking we have to prepare patches. Shortly speaking, the role of patches was played above by lines. The generalization below is a transition from lines to curves. Therefore we proceed with a preliminary study of curves.

1.2. Logarithmic asymptotes of a curve. As is known since Newton’s works (see [New67]), behavior of a curve {(x, y) ∈ R2 | a(x, y)=0} near the coordinate axes and at infinity depends, as a rule, on the coefficients of a corresponding to the boundary points of its Newton polygon ∆(a). We need more specific formulations, but prior to that we have to introduce several notations and discuss some notions. R2 ω1 ω2 For a set Γ ⊂ and a polynomial a(x, y) = ω∈Z2 aωx y , denote the ω1 ω2 Γ polynomial 2 a x y by a . It is called the Γ-truncation of a. ω∈Γ∩Z ω P For a set U ⊂ R2 and a real polynomial a in two variables, denote the curve P {(x, y) ∈ U | a(x, y)=0} by VU (a). The complement of the coordinate axes in R2, i.e. a set {(x, y) ∈ R2 | xy 6= 0}, is denoted3 by RR2.

3This notation is motivated in Section 2.3 below. PATCHWORKING REAL ALGEBRAIC VARIETIES 5

+ +

+ ++ + +

+ +

+ + ++ + + + +

+ + +

+ +

Figure 4. Harnack’s curve of degree 6.

Denote by l the map RR2 → R2 defined by formula l(x, y) = (ln |x|, ln |y|). It is clear that the restriction of l to each quadrant is a diffeomorphism. A polynomial in two variables is said to be quasi-homogeneous if its Newton polygon is a segment. The simplest real quasi-homogeneous polynomials are bino- p q mials of the form αx + βy where p and q are relatively prime. A curve VRR2 (a), where a is a binomial, is called quasiline. The map l transforms quasilines to lines. In that way any line with rational slope can be obtained. The image l(VRR2 (a)) of the quasiline VRR2 (a) is orthogonal to the segment ∆(a). It is clear that any real quasi-homogeneous polynomial in 2 variables is decomposable into a product of binomials of the type described above and trinomials without zeros in RR2. Thus if a is a real quasi-homogeneous polynomial then the curve VRR2(a) is decomposable into a union of several quasilines which are transformed by l to lines orthogonal to ∆(a). A real polynomial a in two variables is said to be peripherally nondegenerate if Γ for any side Γ of its Newton polygon the curve VRR2 (a ) is nonsingular (it is a union of quasilines transformed by l to parallel lines, so the condition that it is nonsingular means absence of multiple components). Being peripherally nondegenerate is typical in the sense that among polynomials with the same Newton polygons the peripherally nondegenerate ones form nonempty set open in the Zarisky topology. − For a side Γ of a polygon ∆, denote by DC∆ (Γ) a ray consisting of vectors orthogonal to Γ and directed outside ∆ with respect to Γ (see Figure 6 and Section 2.2). The assertion in the beginning of this Section about behavior of a curve nearby the coordinate axes and at infinity can be made now more precise in the following way. 2 1.2.A. Let ∆ ∈ RR be a convex polygon with nonempty interior and sides Γ1, ..., Γn. Let a be a peripherally nondegenerate real polynomial in 2 variables with 6 OLEG VIRO

+ +

+ + + + +

+ + +

+ + + ++ + + +

+ + + + + + + +

+ + +

+ +

Figure 5. Gudkov’s curve of degree 6.

Figure 6.

2 Γi ∆(a) = ∆. Then for any quadrant U ∈ RR each line contained in l(VU (a ) with i = 1,. . . ,n is an asymptote of l(VU (a)), and l(VU (a)) goes to inﬁnity only along − these asymptotes in the directions deﬁned by rays DC∆(Γi). PATCHWORKING REAL ALGEBRAIC VARIETIES 7

Figure 7.

Figure 8.

Theorem generalizing this proposition is formulated in Section 6.3 and proved in Section 6.4. Here we restrict ourselves to the following elementary example illustrating 1.2.A. Example 1.2.B. Consider the polynomial a(x, y)=8x3 − x2 +4y2. Its Newton polygon is shown in Figure 6. In Figure 7 the curve VR2 (a) is shown. In Figure − Γi 8 the rays DC∆ (Γi) and the images of VU (a) and VU (a ) under diﬀeomorphisms 2 2 l|U : U → R are shown, where U runs over the set of components of RR (i.e. − quadrants). In Figure 9 the images of DC∆(Γi) under l and the curves VR2 (a) and 2 Γi VR (a ) are shown.

1.3. Charts of polynomials. The notion of a chart of a polynomial is fundamental for what follows. It is introduced naturally via the theory of toric varieties (see 8 OLEG VIRO

Section 3). Another definition, which is less natural and applicable to a narrower class of polynomials, but more elementary, can be extracted from the results generalizing Theorem 1.2.A (see Section 6). In this Section, first, I try to give a rough idea about the definition related with toric varieties, and then I give the definitions related with Theorem 1.2.A with all details. To any convex closed polygon ∆ ⊂ R2 with vertices whose coordinates are in- tegers, a real algebraic surface R∆ is associated. This surface is a completion of RR2 (= (Rr 0)2). The complement R∆ r RR2 consists of lines corresponding to sides of ∆. From the topological viewpoint R∆ can be obtained from four copies of ∆ by pairwise gluing of their sides. For a real polynomial a in two variables we denote the closure of VRR2 (a) in R∆ by VR∆(a). Let a be a real polynomial in two variables which is not quasi-homogeneous. (The latter assumption is not nec- essary, it is made for the sake of simplicity.) Cut the surface R∆(a) along lines of R∆(a)rRR2 (i.e. replace each of these lines by two lines). The result is four copies of ∆(a) and a curve lying in them obtained from VR∆(a)(a). The pair consisting of these four polygons and this curve is a chart of a. Recall that for ε,δ = ±1 we denote the reflection R2 → R2 : (x, y) 7→ (εx,δy) 2 by Sε,δ. For a set A ⊂ R we denote Sε,δ(A) by Aε,δ (see Figure 2). Denote a 2 quadrant {(x, y) ∈ R | εx > 0,δy > 0} by Qε,δ. Now define the charts for two classes of real polynomials separately. First, consider the case of quasi-homogeneous polynomials. Let a be a quasi- homogeneous polynomial defining a nonsingular curve VRR2 (a). Let (w1, w2) be a vector orthogonal to ∆ = ∆(a) with integer relatively prime coordinates. It is clear that in this case VR2 (a) is invariant under S(−1)w1 ,(−1)w2 . A pair (∆∗,υ) consisting of ∆∗ and a finite set υ ⊂ ∆∗ is called a chart of a, if the number of points of

υ ∩ ∆ε,δ is equal to the number of components of VQε,δ (a) and υ is invariant under S(−1)w1 ,(−1)w2 (remind that VR2 (a) is invariant under the same reﬂection).

6 Example 1.3.A. In Figure 10 it is shown a curve VR2 (a) with a(x, y)=2x y − x4y2 − 2x2y3 + y4 = (x2 − y)(x2 + y)(2x2 − y)y, and a chart of a. Now consider the case of peripherally nondegenerate polynomials with Newton polygons having nonempty interiors. Let ∆, Γ1,..., Γn and a be as in 1.2.A. Then, as it follows from 1.2.A, there exist a disk D ⊂ R2 with center at the origin and neighborhoods − − D1,...,Dn of rays DC∆(Γ1),...,DC∆(Γn) such that the curve VRR2 (a) lies in −1 −1 l (D ∪D1 ∪···∪Dn) and for i =1,...,n the curve Vl (DirD)(a) is approximated Γi −1 −1 by Vl (DirD)(a ) and can be contracted (in itself) to Vl (Di∩∂D)(a).

Figure 9. PATCHWORKING REAL ALGEBRAIC VARIETIES 9

Figure 10.

- - - +

Figure 11.

A pair (∆∗, υ) consisting of ∆∗ and a curve υ ⊂ ∆∗ is called a chart of a if Γi (1) for i =1,...,n the pair (Γi∗, Γi∗ ∩ υ) is a chart of a and (2) for ε, δ = ±1 there exists a homeomorphism hε,δ : D → ∆ such that

−1 υ ∩ ∆ε,δ = Sε,δ ◦ hε,δ ◦ l(Vl (D)∩Qε,δ (a)) and hε,δ(∂D ∩ Di) ⊂ Γi for i = 1,...,n. It follows from 1.2.A that any peripherally nondegenerate real polynomial a with Int ∆(a) 6= ∅ has a chart. It is easy to see that the chart is unique up to a homeomorphism ∆∗ → ∆∗ preserving the polygons ∆ε,δ, their sides and their vertices. Example 1.3.B. In Figure 11 it is shown a chart of 8x3 − x2 +4y2 which was considered in 1.2.B.

1.3.C (Generalization of Example 1.3.B). Let

i1 j1 i2 j2 i3 j3 a(x, y)= a1x y + a2x y + a3x y be a non-quasi-homogeneous real polynomial (i. e., a real trinomial whose the New- ton polygon has nonempty interior). For ε,δ = ±1 set

ik jk σεik ,δjk = sign(akε δ ).

Then the pair consisting of ∆∗ and the midlines of ∆ε,δ separating the vertices

(εik, δjk) with opposite signs σεik ,δjk is a chart of a.

Proof. Consider the restriction of a to the quadrant Qε,δ. If all signs σεik ,δjk are the same, then aQε,δ is a sum of three monomials taking values of the same sign on 10 OLEG VIRO

Qε,δ. In this case VQε,δ (a) is empty. Otherwise, consider the side Γ of the triangle ∆ on whose end points the signs coincide. Take a vector (w1, w2) orthogonal to Γ. w1 w2 Consider the curve deﬁned by parametric equation t 7→ (x0t ,y0t ). It is easy to see that the ratio of the monomials corresponding to the end points of Γ does not change along this curve, and hence the sum of them is monotone. The ratio of each of these two monomials with the third one changes from 0 to −∞ monotonically. Therefore the trinomial divided by the monomial which does not sit on Γ changes from −∞ to 1 continuously and monotonically. Therefore it takes the zero value w1 w2 once. Curves t 7→ (x0t ,y0t ) are disjoint and ﬁll Qε,δ. Therefore, the curve

VQε,δ (a) is isotopic to the preimage under Sε,δ ◦hε,δ ◦l of the midline of the triangle ∆ε,δ separating the vertices with opposite signs.

1.3.D. If a is a peripherally nondegenerate real polynomial in two variables then 2 the topology of a curve VRR2 (a) (i.e. the topological type of pair (RR , VRR2 (a))) and the topology of its closure in R2, RP 2 and other toric extensions of RR2 can be recovered from a chart of a.

The part of this proposition concerning to VRR2 (a) follows from 1.2.A. See below 2 Sections 2 and 3 about toric extensions of RR and closures of VRR2 (a) in them. In the next Subsection algorithms recovering the topology of closures of VRR2 (a) in R2 and RP 2 from a chart of a are described.

1.4. Recovering the topology of a curve from a chart of the polynomial. First, I shall describe an auxiliary algorithm which is a block of two main algorithms of this Section. 1.4.A (Algorithm. Adjoining a side with normal vector (α, β)). Initial data: a chart (∆∗, υ) of a polynomial. − If ∆ (= ∆++) has a side Γ with (α, β) ∈ DC∆ (Γ) then the algorithm does not change (∆∗, υ). Otherwise: 1. Drawn the lines of support of ∆ orthogonal to (α, β). 2. Take the point belonging to ∆ on each of the two lines of support, and join these points with a segment. 3. Cut the polygon ∆ along this segment. 4. Move the pieces obtained aside from each other by parallel translations deﬁned by vectors whose diﬀerence is orthogonal to (α, β). 5. Fill the space obtained between the pieces with a parallelogram whose opposite sides are the edges of the cut. 6. Extend the operations applied above to ∆ to ∆∗ using symmetries Sε,δ. 7. Connect the points of edges of the cut obtained from points of υ with segments which are parallel to the other pairs of the sides of the parallelograms inserted, and adjoin these segments to what is obtained from υ. The result and the polygon obtained from ∆∗ constitute the chart produced by the algorithm. Example 1.4.B. In Figure 12 the steps of Algorithm 1.4.A are shown. It is applied to (α, β) = (−1, 0) and the chart of 8x3 − x2 +4y2 shown in Figure 11. Application of Algorithm 1.4.A to a chart of a polynomial a (in the case when it does change the chart) gives rise a chart of polynomial (xβ y−α + x−βyα)x|β|y|α|a(x, y). PATCHWORKING REAL ALGEBRAIC VARIETIES 11

Figure 12.

If ∆ is a segment (i.e. the initial polynomial is quasi-homogeneous) and this segment is not orthogonal to the vector (α, β) then Algorithm 1.4.A gives rise to a chart consisting of four parallelograms, each of which contains as many parallel segments as components of the curve are contained in corresponding quadrant. 1.4.C (Algorithm). Recovering the topology of an affine curve from a chart of the polynomial. Initial data: a chart (∆∗, υ) of a polynomial. 1. Apply Algorithm 1.4.A with (α, β) = (0, −1) to (∆∗, υ). Assign the former notation (α, β) to the result obtained. 2. Apply Algorithm 1.4.A with (α, β) = (0, −1) to (∆∗, υ). Assign the former notation (α, β) to the result obtained. 3. Glue by S+,− the sides of ∆+,δ, ∆−,δ which are faced to each other and parallel to (0, 1) (unless the sides coincide). 4. Glue by S−,+ the sides of ∆ε,+, ∆ε,− which are faced to each other and parallel to (1, 0) (unless the sides coincide). 5. Contract to a point all sides obtained from the sides of ∆ whose normals are directed into quadrant P−,−. 6. Remove the sides which are not touched on in blocks 3, 4 and 5. ′ Algorithm 1.4.C turns the polygon ∆∗ to a space ∆ which is homeomorphic to R2, and the set υ to a set υ′ ⊂ ∆′ such that the pair (∆′, υ′) is homeomorphic 2 to (R , Cl VRR2 (a)), where Cl denotes closure and a is a polynomial whose chart is (∆∗, υ). Example 1.4.D. In Figure 13 the steps of Algorithm 1.4.C applying to a chart of polynomial 8x3y − x2y +4y3 are shown. 12 OLEG VIRO

Figure 13.

1.4.E (Algorithm). Recovering the topology of a projective curve from a chart of the polynomial. Initial data: a chart (∆∗, υ) of a polynomial. 1. Block 1 of Algorithm 1.4.C. 2. Block 2 of Algorithm 1.4.C. 3. Apply Algorithm 1.4.A with (α, β) = (1, 1) to (∆∗, υ). Assign the former notation (∆∗, υ) to the result obtained. 4. Block 3 of Algorithm 1.4.C. 5. Block 4 of Algorithm 1.4.C. 6. Glue by S−,− the sides of ∆++ and ∆−− which are faced to each other and orthogonal to (1, 1). 7. Glue by S−,− the sides of ∆+− and ∆−+ which are faced to each other and orthogonal to (1, −1). 8. Block 5 of Algorithm 1.4.C. 9. Contract to a point all sides obtained from the sides of ∆ with normals directed into the angle {(x, y) ∈ R2 | x< 0, y + x> 0}. 10. Contract to a point all sides obtained from the sides of ∆ with normals directed into the angle {(x, y) ∈ R2 | y< 0, y + x> 0}.

′ Algorithm 1.4.E turns polygon ∆∗ to a space ∆ which is homeomorphic to projective plane RP 2, and the set υ to a set υ′ such that the pair (∆′, υ′) is PATCHWORKING REAL ALGEBRAIC VARIETIES 13

2 homeomorphic to (RP , VRR2 (a)), where a is the polynomial whose chart is the initial pair (∆∗, υ).

1.5. Patchworking charts. Let a1,...,as be peripherally nondegenerate real polynomials in two variables with Int ∆(ai) ∩ Int ∆(aj ) = ∅ for i 6= j. A pair (∆∗, υ) s is said to be obtained by patchworking if ∆ = i=1 ∆(ai) and there exist charts s (∆(ai)∗, υi) of a1,...,as such that υ = υi. i=1 S Example 1.5.A. In Figure 11 and FigureS 14 charts of polynomials 8x3 − x2 +4y2 and 4y2 − x2 + 1 are shown. In Figure 15 the result of patchworking these charts is shown.

Figure 14 Figure 15

1.6. Patchworking polynomials. Let a1,...,as be real polynomials in two vari- ∅ ∆(ai)∩∆(aj ) ∆(ai)∩∆(aj ) ables with Int ∆(ai) ∩ Int ∆(aj ) = for i 6= j and ai = aj for s any i, j. Suppose the set ∆ = i=1 ∆(ai) is convex. Then, obviously, there exists the unique polynomial a with ∆(a) = ∆ and a∆(ai) = a for i =1,...,s. S i Let ν : ∆ → R be a convex function such that:

(1) restrictions ν|∆(ai) are linear; (2) if the restriction of ν to an open set is linear then the set is contained in one of ∆(ai); (3) ν(∆ ∩ Z2) ⊂ Z.

Then ν is said to convexify the partition ∆(a1),..., ∆(as) of ∆. ω1 ω2 If a(x, y)= ω∈Z2 aωx y then we put

ω1 ω2 ν(ω1,ω2) P bt(x, y)= aωx y t Z2 ωX∈ and say that polynomials bt are obtained by patchworking a1,...,as by ν. 3 2 2 2 2 Example 1.6.A. Let a1(x, y)=8x − x +4y , a2(x, y)=4y − x + 1 and

0, if ω1 + ω2 ≥ 2 ν(ω1,ω2)= (2 − ω1 − ω2, if ω1 + ω2 ≤ 2. 3 2 2 2 Then bt(x, y)=8x − x +4y + t . 14 OLEG VIRO

Figure 16.

1.7. The Main Patchwork Theorem. A real polynomial a in two variables is said to be completely nondegenerate if it is peripherally nondegenerate (i.e. for Γ any side Γ of its Newton polygon the curve VRR2 (a ) is nonsingular) and the curve VRR2 (a) is nonsingular.

1.7.A. If a1,...,as are completely nondegenerate polynomials satisfying all conditions of Section 1.6, and bt are obtained from them by patchworking by some nonnegative convex function ν convexifying ∆(a1),..., ∆(as), then there exists t0 > 0 such that for any t ∈ (0,t0] the polynomial bt is completely nondegenerate and its chart is obtained by patchworking charts of a1,...,as. By 1.3.C, Theorem 1.7.A generalizes Theorem 1.1.D. Theorem generalizing The- orem 1.7.A is proven in Section 4.3. Here we restrict ourselves to several examples. Example 1.7.B. Polynomial 8x3 − x2 +4y2 + t2 with t> 0 small enough has the chart shown in Figure 15. See examples 1.5.A and 1.6.A. In the next Section there are a number of considerably more complicated examples demonstrating efficiency of Theorem 1.7.A in the topology of real algebraic curves. 1.8. Construction of M-curves of degree 6. One of central points of the well known 16th Hilbert’s problem [Hil01] is the problem of isotopy classification of curves of degree 6 consisting of 11 components (by the Harnack inequality [Har76] the number of components of a curve of degree 6 is at most 11). Hilbert conjectured that there exist only two isotopy types of such curves. Namely, the types shown in Figure 16 (a) and (b). His conjecture was disproved by Gudkov [GU69] in 1969. Gudkov constructed a curve of degree 6 with ovals’ disposition shown in Figure 16 (c) and completed solution of the problem of isotopy classification of nonsingular curves of degree 6. In particular, he proved, that any curve of degree 6 with 11 components is isotopic to one of the curves of Figure 16. Gudkov proposed twice — in [Gud73] and [Gud71] — simplified proofs of real- izability of the third isotopy type. His constructions, however, are essentially more complicated than the construction described below, which is based on 1.7.A and besides gives rise to realization of the other two types, and, after a small modification, realization of almost all isotopy types of nonsingular plane projective real algebraic curves of degree 6 (see [Vir89]). Construction In Figure 17 two curves of degree 6 are shown. Each of them has one singular point at which three nonsingular branches are second order tangent PATCHWORKING REAL ALGEBRAIC VARIETIES 15

Figure 17.

to each other (i.e. this singularity belongs to type J10 in the Arnold classification [AVGZ82]). The curves of Figure 17 (a) and (b) are easily constructed by the Hilbert method [Hil91], see in [Vir89], Section 4.2. Choosing in the projective plane various affine coordinate systems, one obtains various polynomials defining these curves. In Figures 18 and 19 charts of four polynomials appeared in this way are shown. In Figure 20 the results of patchworking charts of Figures 18 and 19 are shown. All constructions can be done in such a way that Theorem 1.7.A (see [Vir89], Section 4.2) may be applied to the corresponding polynomials. It ensures existence of polynomials with charts shown in Figure 20.

1.9. Behavior of curve VRR2 (bt) as t → 0. Let a1,...,as, ∆ and ν be as in Section 1.6. Suppose that polynomials a1,...,as are completely nondegenerate and ν|∆(a1) = 0. According to Theorem 1.7.A, the polynomial bt with suﬃciently small t > 0 has a chart obtained by patchworking charts of a1,...,as. Obviously, b0 = a1 since ν|∆(a1) = 0. Thus when t comes to zero the chart of a1 stays only, the other charts disappear. How do the domains containing the pieces of VRR2 (bt) homeomorphic to VRR2 (a1), ..., VRR2 (as) behave when t approaches zero? They are moving to the coordinate axes and inﬁnity. The closer t to zero, the more place is occupied by the domain, where VRR2 (bt) is organized as VRR2 (a1) and is approximated by it (cf. Section 6.7). It is curious that the family bt can be changed by a simple geometric transformation in such a way that the role of a1 passes to any one of a2,...,as or even Γ R2 R to ak , where Γ is a side of ∆(ak), k = 1,...,s. Indeed, let λ : → be a ′ ′ linear function, λ(x, y) = αx + βy + γ. Let ν = ν − λ. Denote by bt the re- ′ sult of patchworking a1,...,as by ν . Denote by qh(a,b),t the linear transformation RR2 → RR2 : (x, y) 7→ (xta,ytb). Then ′ VRR2 (bt)= VRR2 (bt ◦ qh(−α,−β),t)= qh(α,β),tVRR2 (bt). Indeed,

′ ω1 ω2 ν(ω1,ω2)−αω1−βω2−γ bt(x, y)= aωx y t

−γ −α ω1 −β ω2 ν(ω1,ω2) X= t aω(xt ) (yt ) t −γ −α −β = t bXt(xt ,yt ) −γ = t bt ◦ qh(−α,−β),t(x, y). 16 OLEG VIRO

Figure 18

Figure 19 Figure 20

′ Thus the curves VRR2 (bt) and VRR2 (bt) are transformed to each other by a linear ′ transformation. However the polynomial bt does not tend to a1 as t → 0. For ′ ′ example, if λ|∆(ak) = ν|∆(ak) then ν |∆(ak) = 0 and bt → ak. In this case as t → 0, ′ the domains containing parts of VRR2 (bt), which are homeomorphic to VRR2 (ai), ′ with i 6= k, run away and the domain in which VRR2 (bt) looks like VRR2 (ak) occupies more and more place. If the set, where ν coincides with λ (or diﬀers from λ by ′ Γ a constant), is a side Γ of ∆(ak), then the curve VRR2 (bt) turns to VRR2 (ak ) (i.e. collection of quasilines) as t → 0 similarly. The whole picture of evolution of VRR2 (bt) when t → 0 is the following. The fragments which look as VRR2 (ai) with i =1,...,s become more and more explicit, but these fragments are not staying. Each of them is moving away from the others. The only fragment that is growing without moving corresponds to the set where ν is constant. The other fragments are moving away from it. From the metric viewpoint some of them (namely, ones going to the origin and axes) are contracting, while the others are growing. But in the logarithmic coordinates, i.e. being transformed by PATCHWORKING REAL ALGEBRAIC VARIETIES 17

Figure 21.

Figure 22.

l : (x, y) 7→ (ln |x|, ln |y|), all the fragments are growing (see Section 6.7). Changing ν we are applying linear transformation, which distinguishes one fragment and casts away the others. The transformation turns our attention to a new piece of the curve. It is as if we would transfer a magnifying lens from one fragment of the curve to another. Naturally, under such a magniﬁcation the other fragments disappear at the moment t = 0.

1.10. Patchworking as smoothing of singularities. In the projective plane the passage from curves defined by bt with t> 0 to the curve defined by b0 looks quite differently. Here, the domains, in which the curve defined by bt looks like curves defined by a1,...,as are not running away, but pressing more closely to the points (1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1) and to the axes joining them. At t = 0, they are pressed into the points and axes. It means that under the inverse passage (from t = 0 to t > 0) the full or partial smoothing of singularities concentrated at the points (1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1) and along coordinate axes happens.

∆(a1)∩∆(a2) Example 1.10.A. Let a1, a2 be polynomials of degree 6 with a1 = ∆(a1)∩∆(a2) a2 and charts shown in Figure 18 (a) and 19 (b). Let ν1, ν2 and ν3 18 OLEG VIRO be deﬁned by the following formulas:

0, if ω1 +2ω2 ≤ 6 ν1(ω1,ω2)= (2(ω1 +2ω2 − 6), if ω1 +2ω2 ≥ 6

6 − ω1 − 2ω2, if ω1 +2ω2 ≤ 6 ν2(ω1,ω2)= (ω1 +2ω2 − 6, if ω1 +2ω2 ≥ 6

2(6 − ω1 − 2ω2), if ω1 +2ω2 ≤ 6 ν3(ω1,ω2)= (0, if ω1 +2ω2 ≥ 6 1 (note, that ν1, ν2 and ν3 differ from each other by a linear function). Let bt , 2 3 bt and bt be the results of patchworking a1, a2 by ν1, ν2 and ν3. By Theorem 1 2 3 1.7.A for sufficiently small t> 0 the polynomials bt , bt and bt have the same chart shown in Figure 20 (ab), but as t → 0 they go to different polynomials, namely, a1, ∆(a1)∩∆(a2) i a1 and a2.The closure of VRR2 (bt) with i = 1, 2, 3 in the projective plane (they are transformed to one another by projective transformations) are shown in Figure 21. The limiting projective curves, i.e. the projective closures of VRR2 (a1), ∆(a1)∩∆(a2) VRR2 (a1 ), VRR2 (a2) are shown in Figure 22. The curve shown in Figure 22 (b) is the union of three nonsingular conics which are tangent to each other in two points. Curves of degree 6 with eleven components of all three isotopy types can be obtained from this curve by small perturbations of the type under consideration (cf. Section 1.8). Moreover, as it is proven in [Vir89], Section 5.1, nonsingular curves of degree 6 of almost all isotopy types can be obtained.

1.11. Evolvings of singularities. Let f be a real polynomial in two variables. (See Section 5, where more general situation with an analytic function playing the role of f is considered.) Suppose its Newton polygon ∆(f) intersects both coordinate axes (this assumption is equivalent to the assumption that VR2 (f) is the closure of VRR2 (f)). Let the distance from the origin to ∆(f) be more than 1 or, equivalently, the curve VR2 (f) has a singularity at the origin. Let this singularity be isolated. Denote by B a disk with the center at the origin having suﬃciently small radius such that the pair (B, VB (f)) is homeomorphic to the cone over its boundary (∂B,V∂B(f)) and the curve VR2 (f) is transversal to ∂B (see [Mil68], Theorem 2.10). Let f be included into a continuous family ft of polynomials in two variables: f = f0. Such a family is called a perturbation of f. We shall be interested mainly in perturbations for which curves VR2 (ft) have no singular points in B when t is in some segment of type (0,ε]. One says about such a perturbation that it evolves the singularity of VR2 (ft) at zero. If perturbation ft evolves the singularity of VR2 (f) at zero then one can ﬁnd t0 > 0 such that for t ∈ (0,t0] the curve VR2 (ft) has no singularities in B and, moreover, is transversal to ∂B. Obviously, there exists an isotopy ht : B → B with t0 ∈ (0,t0] such that ht0 = id and ht(VB(f0)) = VB (ft), so all pairs (B, VB (ft)) with t ∈ (0,t0] are homeomorphic to each other. A family (B, VR2 (ft)) of pairs with t ∈ (0,t0] is called an evolving of singularity of VR2 (f) at zero, or an evolving of germ of VR2 (f). Denote by Γ1,..., Γn the sides of Newton polygon ∆(f) of the polynomial f, n faced to the origin. Their union Γ(f) = i=1 Γi is called the Newton diagram of f. S PATCHWORKING REAL ALGEBRAIC VARIETIES 19

Γi Suppose the curves VRR2 (f ) with i =1,...,n are nonsingular. Then, according Γi to Newton [New67], the curve VR2 (f) is approximated by the union of Cl VRR2 (f ) with i =1,...,n in a sufficiently small neighborhood of the origin. (This is a local version of Theorem 1.2.A; it is, as well as 1.2.A, a corollary of Theorem 6.3.A.) Disk Γi B can be taken so small that V∂B(f) is close to ∂B ∩ VRR2 (f ), so the number Γ and disposition of these points are defined by charts (Γi∗, υi) of fi . The union n n (Γ(f)∗, υ) = ( i=1 Γi∗, i=1 υi) of these charts is called a chart of germ of VR2 (f) at zero. It is a pair consisting of a simple closed polygon Γ(f∗), which is symmetric with respect toS the axes andS encloses the origin, and finite set υ lying on it. There is a natural bijection of this set to V∂B(f), which is extendable to a homeomorphism (Γ(f)∗, υ) → (∂B,V∂B (f)). Denote this homeomorphism by g. Let ft be a perturbation of f, which evolves the singularity at the origin. Let B, t0 and ht be as above. It is not difficult to choose an isotopy ht : B → B, t ∈ (0,t0] ′ such that its restriction to ∂B can be extended to an isotopy ht : ∂B → ∂B with ′ t ∈ [0,t0] and h0(V∂B(ft0 )) = V∂B (f). A pair (Π, τ) consisting of the polygon Π bounded by Γ(f)∗ and an 1-dimensional subvariety τ of Π is called a chart of evolving (B, VB (ft)), t ∈ [0,t0] if there exists a homeomorphism Π → B, mapping

τ to V∂B (ft0 )∗, whose restriction ∂Π → ∂Π is the composition Γ(f)∗@ > g >> ′ ∂B@ > h0 >> ∂B. It is clear that the boundary (∂Π, ∂τ) of a chart of germ’s evolving is a chart of the germ. Also it is clear that if polynomial f is completely nondegenerate and polygons ∆(ft) are obtained from ∆(f) by adjoining the region restricted by the axes and Π(f), then charts of ft with t ∈ (0,t0] can be obtained by patchworking a chart of f and chart of evolving (B, VB (ft)), t ∈ [0,t0]. The patchworking construction for polynomials gives a wide class of evolvings whose charts can be created by the modiﬁcation of Theorem 1.7.A formulated below. Let a1,...,as be completely nondegenerate polynomials in two variables with ∅ ∆(ai)∩∆(aj ) ∆(ai)∩∆(aj ) s Int ∆(ai)∩Int ∆(aj )= and ai = aj for i 6= j. Let i=1 ∆(ai) ∆(ai)∩∆(f) ∆(ai)∩∆(f) be a polygon bounded by the axes and Γ(f). Let ai = f S for i = a,...,s. Let ν : R2 → R be a nonnegative convex function which is equal to s zero on ∆(f) and whose restriction on i=1 ∆(ai) satisﬁes the conditions 1, 2 and 3 of Section 1.6 with respect to a1,...,as. Then a result ft of patchworking f, S a1,...,as by ν is a perturbation of f. Theorem 1.7.A cannot be applied in this situation because the polynomial f is not supposed to be completely nondegenerate. This weakening of assumption implies a weakening of conclusion.

1.11.A (Local version of Theorem 1.7.A). Under the conditions above perturbation ft of f evolves a singularity of VR2 (f) at the origin. A chart of the evolving can be obtained by patchworking charts of a1,...,as. An evolving of a germ, constructed along the scheme above, is called a patchwork evolving. Γ(f) If Γ(f) consists of one segment and the curve VRR2 (f ) is nonsingular then the germ of VR2 (f) at zero is said to be semi-quasi-homogeneous. In this case for construction of evolving of the germ of VR2 (f) according the scheme above we need only one polynomial; by 1.11.A, its chart is a chart of evolving. In this case geometric structure of VB (ft) is especially simple, too: the curve VB (ft) is approximated by qhw,t(VR2 (a1)), where w is a vector orthogonal to Γ(f), that is 20 OLEG VIRO by the curve VR2 (a1) contracted by the quasihomothety qhw,t. Such evolvings were described in my paper [Vir80]. It is clear that any patchwork evolving of semi- quasi-homogeneous germ can be replaced, without changing its topological models, by a patchwork evolving, in which only one polynomial is involved (i.e. s = 1). PATCHWORKING REAL ALGEBRAIC VARIETIES 21

2. Toric varieties and their hypersurfaces 2.1. Algebraic tori KRn. In the rest of this chapter K denotes the main field, which is either the real number field R, or the complex number field C. n For ω = (ω1,...,ωn) ∈ Z and ordered collection x of variables x1,...,xn the ω1 ωn ω product x1 ...xn is denoted by x . A linear combination of products of this sort with coefficients from K is called a Laurent polynomial or, briefly, L-polynomial over −1 −1 K. Laurent polynomials over K in n variables form a ring K[x1, x1 ,...,xn, xn ] naturally isomorphic to the ring of regular functions of the variety (K r 0)n. Below this variety, side by side with the affine space Kn and the projective space KP n, is one of the main places of action. It is an algebraic torus over K. Denote it by KRn. n n Denote by l the map KR → R defined by formula l(x1,...,xn) = (ln |x1|,..., ln |xn|). 0 1 Put UK = {x ∈ K | |x| = 1}, so UR = S and UC = S . Denote by ar the map Rn n x1 xn K → UK (= UK ×···× UK ) defined by ar(x1,...,xn) = ( ,..., ). |x1| |xn| Denote by la the map Rn Rn n x 7→ (l(x), ar(x)) : K → × UK . It is clear that this is a diffeomorphism. KRn is a group with respect to the coordinate-wise multiplication, and l, ar, la are group homomorphisms; la is an isomorphism of KRn to the direct product of Rn n (additive) group and (multiplicative) group UK. Being Abelian group, KRn acts on itself by translations. Let us fix notations for some of the translations involved into this action. n For w ∈ R and t> 0 denote by qhw,t and call a quasi-homothety with weights n n w = (w1,...,wn) and coefficient t the transformation KR → KR defined by for- w1 wn w1 wn mula qhw,t(x1,...,xn) = (t x1,...,t xn), i.e. the translation by (t ,...,t ). If w = (1,..., 1) then it is the usual homothety with coefficient t. It is clear that qhw,t = qhλ−1w,t for λ > 0. Denote by qhw a quasi-homothety qhw,e, where e is the base of natural logarithms. It is clear, qhw,t = qh(ln t)w. n Rn Rn For w = (w1,...,wn) ∈ UK denote by Sw the translation K → K defined by formula Sw(x1,...,xn) = (w1x1,...,wnxn), i. e. the translation by w. n n n For w ∈ R denote by Tw the translation x 7→ x + w : R → R by the vector w. n n n 2.1.A. R R n Diffeomorphism la : K → × UK transforms qhw,t to T(ln t)w × idUK , Rn n and Sw to id ×(Sw|UK ), i.e. −1 la ◦ qh ◦ la = T × id n and w,t (ln t)w UK −1 Rn n la ◦ Sw ◦ la = id ×(Sw|UK ).

−1 In particular, la ◦ qhw ◦ la = Tw × id. A hypersurface of KRn deﬁned by a(x) = 0, where a is a Laurent polynomial over K in n variables is denoted by VKRn (a). ω If a(x) = ω∈Zn aωx is a Laurent polynomial, then by its Newton polyhedron n ∆(a) is the convex hull of {ω ∈ R | aω 6=0}. P 22 OLEG VIRO

2.1.B. Let a be a Laurent polynomial over K. If ∆(a) lies in an aﬃne subspace n n Γ of R then for any vector w ∈ R orthogonal to Γ, a hypersurface VKRn (a) is invariant under qhw,t. Proof. Since ∆(a) ⊂ Γ and Γ ⊥ w, then for ω ∈ ∆(a) the scalar product wω does not depend on ω. Hence −1 −w ω −wω ω −wω a(qhw,t(x)) = aω(t x) = t aωx = t a(x), ω∈X∆(a) ω∈X∆(a) and therefore −1 −wω qhw,t(VKRn (a)) = VKRn (a ◦ qhw,t)= VKRn (t a)= VKRn (a).

Proposition 2.1.B is equivalent, as it follows from 2.1.A, to the assertion that under hypothesis of 2.1.B the set la(VKRn (a)) contains together with each point Rn n ′ Rn n ′ (x, y) ∈ × UK all points (x ,y) ∈ × UK with x − x ⊥ Γ. In other words, in n the case ∆(a) ⊂ Γ the intersections of la(VKRn (a)) with ﬁbers R × y are cylinders, whose generators are aﬃne spaces of dimension n − dim Γ orthogonal to Γ. The following proposition can be proven similarly to 2.1.B.

2.1.C. Under the hypothesis of 2.1.B a hypersurface VKRn (a) is invariant under transformations S(eπiw1 ,...,eπiwn ), where w ⊥ Γ, Zn, if K = R w ∈ n (R , if K = C.

In other words, under the hypothesis of 2.1.B the hypersurface VKRn (a) contains together with each its point (x1,...,xn): w1 wn n (1) points ((−1) x1,..., (−1) xn) with w ∈ Z , w ⊥ Γ, if K = R, iw1 iwn n (2) points (e x1,...,e xn) with w ∈ R , w ⊥ Γ, if K = C. 2.2. Polyhedra and cones. Below by a polyhedron we mean closed convex polyhedron lying in Rn, which are not necessarily bounded, but have a finite number of faces. A polyhedron is said to be integer if on each of its faces there are enough points with integer coordinates to define the minimal affine space containing this face. All polyhedra considered below are assumed to be integer, unless the contrary is stated. The set of faces of a polyhedron ∆ is denoted by G(∆), the set of its k-dimensional ′ faces by Gk(∆), the set of all its proper faces by G (∆). By a halfspace of vector space V we will mean the preimage of the closed halfline R+(= {x ∈ R : x ≥ 0}) under a non-zero linear functional V → R (so the boundary hyperplane of a halfspace passes necessarily through the origin). By a cone it is called an intersection of a finite collection of halfspaces of Rn. A cone is a polyhedron (not necessarily integer), hence all notions and notations concerning polyhedra are applicable to cones. The minimal face of a cone is the maximal vector subspace contained in the cone. It is called a ridge of the cone. n For v1,...,vk ∈ R denote by hv1,...,vki the minimal cone containing v1,..., vk; it is called the cone generated by v1,...,vk. A cone is said to be simplicial PATCHWORKING REAL ALGEBRAIC VARIETIES 23 if it is generated by a collection of linear independent vectors, and simple if it is generated by a collection of integer vectors, which is a basis of the free Abelian group of integer vectors lying in the minimal vector space which contains the cone. Let ∆ ⊂ Rn be a polyhedron and Γ its face. Denote by C (Γ) the cone r · ∆ r∈R+ (∆−y), where y is a point of Γr∂Γ. The cone C (∆) is clearly the vector subspace ∆ S of Rn which corresponds to the minimal affine subspace containing ∆. The cone CΓ(Γ) is the ridge of C∆(Γ). If Γ is a face of ∆ with dim Γ = dim ∆ − 1, then C∆(Γ) is a halfspace of C∆(∆) with boundary parallel to Γ. For cone C ⊂ Rn we put D+C = {x ∈ Rn | ∀a ∈ C ax ≥ 0}, D−C = {x ∈ Rn | ∀a ∈ C ax ≤ 0}. These are cones, which are said to be dual to C. The cones D+C and D−C are symmetric to each other with respect to 0. The cone D−C permits also the following more geometric description. Each hyperplane of support of C defines a ray consisting of vectors orthogonal to this hyperplane and directed to that of two open halfspaces bounded by it, which does not intersect C. The union of all such rays is D−C. + + − − n It is clear that D D C = C = D D C. If v1,...,vn is a basis of R , then + ∗ ∗ the cone D hv1,...,vni is generated by dual basis v1 ,...,vn (which is defined by conditions vi · vj ∗ = ∆ij ). 2.3. Affine toric variety. Let ∆ ⊂ Rn be an (integer) cone. Consider the semigroup K-algebra K[∆ ∩ Zn] of the semigroup ∆ ∩ Zn. It consists of Laurent poly- ω nomials of the form ω∈∆∩Zn aωx . According to the well known Gordan Lemma (see, for example, [Dan78], 1.3), the semigroup ∆ ∩ Zn is generated by a finite number of elementsP and therefore the algebra K[∆ ∩ Zn] is generated by a finite number of monomials. If this number is greater than the dimension of ∆, then there are nontrivial relations among the generators; the number of relations of minimal generated collection is equal to the difference between the number of generators and the dimension of ∆. An affine toric variety K∆ is the affine scheme Spec K[∆∩Zn]. Its less invariant, but more elementary definition looks as follows. Let p p p p

{α1,...,αp | u1,iαi = vi,1αi,..., up−n,iαi = vp−n,iαi} i=1 i=1 i=1 i=1 X n X X X be a presentation of ∆ ∩ Z by generators and relations (here uij and vij are nonnegative); then the variety K∆ is isomorphic to the aﬃne subvariety of Kp deﬁned by the system

u11 u1p v11 v1p y1 ...yp = y1 ...yp ......  u − v −  p n,1 up−n,p p n,1 vp−n,p  y1 ...yp = y1 ...yp . Rn −1 −1 For example, if ∆ =  , then K∆ = Spec K[x1, x1 ,...,xn, xn ] can be presented as the subvariety of K2n deﬁned by the system

y1yn+1 =1  ......  yny2n =1

 24 OLEG VIRO

Figure 23.

Projection K2n → Kn induces an isomorphism of this subvariety to (K r 0)n = KRn. This explains the notation KRn introduced above. n n If ∆ is the positive orthant A = {x ∈ R | x1 ≥ 0,...,xn ≥ 0}, then K∆ is isomorphic to the affine space Kn. The same takes place for any simple cone. If cone is not simple, then corresponding toric variety is necessarily singular. For example, the angle shown in Figure 23 corresponds to the cone defined in K3 by xy = z2. Let a cone ∆1 lie in a cone ∆2. Then the inclusion in : ∆1 → ∆2 defines an n n inclusion K[∆1 ∩ Z ] ֒→ K[∆2 ∩ Z ] which, in turn, defines a regular map ∗ n n in : Spec K[∆2 ∩ Z ] → Spec K[∆1 ∩ Z ], ∗ i.e. a regular map in : K∆2 → K∆1. The latter can be described in terms of subvarieties of affine spaces in the following way. The formulas, defining coordinates of point in∗(y) as functions of coordinates of y, are the multiplicative versions n of formulas, defining generators of semigroup ∆1 ∩ Z as linear combinations of n generators of the ambient semigroup ∆2 ∩ Z . dim∆ In particular, for any ∆ there is a regular map of KC∆(∆) =∼ KR to K∆. It is not difficult to prove that it is an open embedding with dense image, thus K∆ can be considered as a completion of KRdim∆. An action of algebraic torus KC∆(∆) in itself by translations is extended to its action in K∆. This extension can be obtained, for example, in the following way. Note first, that for defining an action in K∆ it is sufficient to define an action in the ring K[∆ ∩ Zn]. Define an action of KRn on monomials xω ∈ K[δ ∩ Zn] by formula ω ω1 ωn Zn (α1,...,αn)x = α1 ...αn and extend it to the whole ring K[∆∩ ] by linearity. Further, note that if V ⊂ Rn is a vector space, then the map in∗ : KRn → KV is a ∗ n group homomorphism. Elements of kernel of in : KR → KC∆(∆) act identically in K[∆ ∩ Zn]. It allows to extract from the action of KRn in K∆ an action of ∆ KC (∆) in K∆, which extends the action of KC∆(∆) in itself by translations. With each face Γ of a cone ∆ one associates (as with a smaller cone) a variety KΓ ∗ and a map in : K∆ → KΓ. On the other hand there exists a map in∗ : KΓ → K∆ ∗ for which in ◦ in∗ is the identity map KΓ → KΓ. Therefore, in∗ is an embedding whose image is a retract of K∆. From the viewpoint of schemes the map in∗ should be defined by the homomorphism K[∆ ∩ Zn] → K[Γ ∩ Zn] which maps a ω ω Laurent polynomial ω∈∆∩Zn aωx to its Γ-truncation ω∈Γ∩Zn aωx . In terms of subvarieties of affine space, KΓ is the intersection of K∆ with the subspace P P PATCHWORKING REAL ALGEBRAIC VARIETIES 25

Figure 24.

yi1 = yi2 = ··· = yis = 0, where yi1 ,...,yis are the coordinates corresponding to generators of semigroup ∆ ∩ Zn which do not lie in Γ. ∗ Varieties in∗(KΓ) with Γ ∈ Gdim∆−1(∆) cover K∆ r in (KC∆(∆)). Images of algebraic tori KCΓ(Γ) with Γ ∈ G(∆) under the composition ∗ ∗ in in KCΓ(Γ) −−−−→ KΓ −−−−→ K∆ of embeddings form a partition of K∆, which is a smooth stratiﬁcation of K∆. ∗ Closure of the stratum in∗ in (KCΓ(Γ)) in K∆ is in∗(KΓ). Below in the cases when it does not lead to confusion we shall identify KΓ with in∗ KΓ and KCΓ(Γ) ∗ with in∗ in KCΓ(Γ) (i.e. we shall consider KΓ and KCΓ(Γ) as lying in K∆). 2.4. Quasi-projective toric variety. Let ∆ ⊂ Rn be a polyhedron. If Γ is its face and Σ is a face of Γ, then CΓ(Σ) is a face of C∆(Γ) parallel to Γ, and

CC∆(Σ)(CΓ(Σ)) = C∆(Γ), see Figure 24. In particular, C∆(Σ) ⊂ C∆(Γ) and, ∗ hence, the map in : KC∆(Γ) → KC∆(Σ) is defined. It is easy to see that this is an open embedding. Let us glue all KC∆(Γ) with Γ ∈ G(∆) together by these embeddings. The result is denoted by K∆ and called the toric variety associated with ∆. This definition agrees with the corresponding definition from the previous Section: if ∆ is a cone and Σ is its ridge then C∆(Σ) = ∆ and, since the ridge is the minimal face, all KC∆(Γ) with Γ ∈ G(∆) are embedded in KC∆(Σ) and the gluing gives KC∆(Σ) = K∆. For any polyhedron ∆ the toric variety K∆ is quasi-projective. If ∆ is bounded, it is projective (see [GK73] and [Dan78]). A polyhedron ∆ ⊂ Rn is said to be permissible if dim ∆ = n, each face of ∆ has a vertex and for any vertex Γ ∈ G0(∆) the cone C∆(Γ) is simple. If polyhedron ∆ is permissible then variety K∆ is nonsingular and it can be obtained by gluing affine spaces KC∆(Γ) with Γ ∈ G0(∆). The gluing allows the following description. Let n us associate with each cone C∆(Γ) where Γ ∈ G0(∆) an automorphism fΓ : KR → n KR : if C∆(Γ) = hv1,...,vni and vi = (vi1,...,vin) for i =1,...,n, then we put v11 v1n vn1 vnn fΓ(x1,...,xn) = (x1 ...xn ,...,x1 ...xn ). The variety K∆ is obtained by Rn n fΓ gluing to K copies of K by maps KRn −−−−→ KRn ֒→ Kn for all vertices Γ of ∆. (Cf. Khovansky [Kho77].) The variety K∆ is defined by ∆, but does not define it. Indeed, if ∆1 and ∆2 are polyhedra such that there exists a bijection G(∆1) → G(∆2), preserving 26 OLEG VIRO

Figure 25.

dimensions and inclusions and assigning to each face of ∆1 a parallel face of ∆2, then K∆1 = K∆2. Denote by P n the simplex of dimension n with vertices (0, 0,..., 0), (1, 0,..., 0), (0, 1, 0,..., 0),..., (0, 0,..., 1). It is permissible polyhedron. KP n is the n-dimensional projective space (this agrees with its usual notation). 2 Evidently, K(∆1 ×∆2)= K∆1 ×K∆2. In particular, if ∆ ⊂ R is a square with vertices (0, 0), (1, 0), (0, 1) and (1, 1), i.e. if ∆ = P 1 × P 1, then K∆ is a surface isomorphic to nonsingular projective surface of degree 2 (to hyperboloid in the case of K = R2). Polyhedra shown in Figure 25 define the following surfaces: K∆1 is the affine plane with a point blown up; K∆2 is projective plane with a point blown up (R∆2 is 1 the Klein bottle); K∆3 is the linear surface over KP , defined by sheaf O + O(−2) (R∆3 is homeomorphic to torus). dim∆ The variety KC∆(∆) is isomorphic to KR , open and dense in K∆, so K∆ dim∆ can be considered as a completion of KR . Actions of KC∆(∆) in affine parts KC∆(Γ) of K∆ correspond to each other and define an action in K∆ which is an extension of the action of KC∆(∆) in itself by translations. Transformations of K∆ extending qhw,t and Sw are denoted by the same symbols qhw,t and Sw. The complement K∆ r KC∆(∆) is covered by KΣ with Σ ∈ G(C∆(Γ)), Γ ∈ ′ ′ G (∆) or, equivalently, by varieties KCΓ(Σ) with Σ ∈ G(Γ), Γ ∈ G (∆). They ′ comprise varieties KΓ with Γ ∈ G (∆), which also cover K∆ r KC∆(∆). The varieties KΓ are situated with respect to each other in the same manner as the corresponding faces in the polyhedron: K(Γ1 ∩ Γ2) = KΓ1 ∩ KΓ2. Algebraic tori r KCΓ(Γ) = KΓ Σ∈G′(Γ) KΣ form partition of K∆, which is a smooth stratifica- tion; they are orbits of the action of KC (∆) in K∆. S ∆ We shall say that a polyhedron ∆2 is richer than a polyhedron ∆1 if for any face

Γ2 ∈ G(∆2) there exists a face Γ1 ∈ G(∆1) such that C∆2 (Γ2) ⊃ C∆1 (Γ1) (such a face Γ1 is automatically unique), and for each face Γ1 ∈ G(∆1) the cone C∆1 (Γ1) can be presented as the intersection of several cones C∆2 (Γ2) with Γ1 ∈ G(∆2). This definition allows a convenient reformulation in terms of dual cones: a polyhedron + ∆2 is richer than polyhedron ∆1 iff the cones D C∆2 (Γ2) with Γ2 ∈ G(∆2) cover + the set, which is covered by D C∆1 (Γ1) with Γ1 ∈ G(∆1), and the first covering is a refinement of the second.

(Let a polyhedron ∆2 be richer than ∆1. Then the inclusions C∆1 (Γ1) ֒→ C∆2 (Γ2 ∗ in deﬁne for any Γ2 ∈ G(∆2) a regular map . KC∆2 (Γ2) −−−−→ KC∆1 (Γ1) ֒→ K∆1 PATCHWORKING REAL ALGEBRAIC VARIETIES 27

Obviously, these maps commute with the embeddings, by which K∆2 and K∆1 are glued from aﬃne pieces, thus a regular map K∆2 → K∆1 appears. One can show (see, for example, [GK73]) that for any polyhedron ∆1 there exists a richer polyhedron ∆2, deﬁning a nonsingular toric variety K∆2. Such a polyhedron is called a resolution of ∆1 (because it gives a resolution of singularities n of K∆1). If dim ∆ = n (= the dimension of the ambient space R ), then a resolution of ∆ can be found among permissible polyhedra.

2.5. Hypersurfaces of toric varieties. Let ∆ ⊂ Rn be a polyhedron and a be a Laurent polynomial over K in n variables. Let C∆(a)(∆(a)) ⊂ C∆(∆). Then there ω ω exists a monomial x such that ∆(x a) ⊂ C∆(∆). The hypersurface VKC∆(∆) ω does not depend on the choice of x and is denoted simply by VKC∆(∆)(a). Its 4 closure in K∆ is denoted by VK∆(a). Thus, to any Laurent polynomial a over K with C∆(a)(∆(a)) ⊂ C∆(∆), a hypersurface VK∆(a) of K∆ is related. For ω Rn Laurent polynomial a(x) = ω∈Zn aωx and a set Γ ⊂ a Laurent polynomial ω Γ a(x)= n a x is denoted by a and called the Γ-truncation of a. ω∈Γ∩Z ω P 2.5.A. PLet ∆ ⊂ Rn be a polyhedron and a be a Laurent polynomial over K with ′ ′ C∆(a)(∆(a)) ⊂ C∆(∆). If Γ1 ∈ G (∆(a)), Γ2 ∈ G (∆) and C∆(a)(Γ1) ⊂ C∆(Γ2) Γ1 then KΓ2 ∩ VK∆(a)= VKΓ2 (a ).

Proof. Consider KC∆(Γ2). It is a dense subset of KΓ2. Since C∆(a)(Γ1) ⊂ C∆(Γ2), ω ω there exists a monomial x such that ∆(x a) lies in C∆(Γ2) and intersects its ridge exactly in the face obtained from Γ1. Since on KΓ2 ∩ KC∆(Γ2) all monomials, whose exponents do not lie on ridge CΓ2 (Γ2) of C∆(Γ2), equal zero, it follows ω that the intersection {x ∈ KC∆(Γ2) | x a(x)=0} ∩ KΓ2 coincides with {x ∈ ω CΓ (Γ2) KC∆(Γ2) | [x a] 2 (x)=0} ∩ KΓ2. Note ﬁnally, that the latter coincides with VKΓ2 (a1).

2.5.B. Let ∆ and a be as in 2.5.A and Γ2 be a proper face of the polyhedron ∆. If ′ there is no face Γ1 ∈ G (∆(a)) with C∆(a)(Γ1) ⊂ C∆(Γ2) then KΓ2 ⊂ VK∆(a). The proof is analogous to the proof of the previous statement. Denote by SVKRn (a) the set of singular points of VKRn (a), i.e. a set VKRn (a) ∩ n ∂a V Rn ( ). i=1 K ∂xi A Laurent polynomial a is said to be completely nondegenerate (over K) if, for T Γ Γ any face Γ of its Newton polyhedron, SVKRn (a ) is empty and, hence, VKRn (a ) is a nonsingular hypersurface. A Laurent polynomial a is said to be peripherally Γ nondegenerate if for any proper face Γ of its Newton polyhedron SVKRn (a )= ∅. It is not diﬃcult to prove that completely nondegenerate L-polynomials form Zarisky open subset of the space of L-polynomials over K with a given Newton polyhedron, and the same holds true also for peripherally nondegenerate L-polynomials. 2.5.C. If a Laurent polynomial a over K is completely nondegenerate and ∆ ⊂ Rn is a resolution of its Newton polyhedron ∆(a) then the variety VK∆(a) is nonsingular and transversal to all KΓ with Γ ∈ G′(∆). See, for example, [Kho77].

4Here it is meant the closure of K∆ in the Zarisky topology; in the case of K = C the classic topology gives the same result, but in the case of K = R the usual closure may be a nonalgebraic set. 28 OLEG VIRO

Theorem 2.5.C allows various generalizations related with possibilities to consider singular K∆ or only some faces of ∆(a) (instead of all of them). For example, one can show that if under the hypothesis of 2.5.A a truncation aΓ of a is completely nondegenerate then under an appropriate understanding of transversality (in the sense of stratiﬁed space theory) VK∆(a) is transversal to KΓ2. Without going into discussion of transversality in this situation, I formulate a special case of this proposition, generalizing Theorem 2.5.C. n 2.5.D. Let Γ be a face of a polyhedron ∆ ⊂ R with nonempty G0(Γ) and with simple cones C∆(Σ) for all Σ ∈ G0(Γ). Let a be a Laurent polynomial over K in Γ n variables and Γ1 be a face of ∆(a) with C∆(a)(Γ1) ⊂ C∆(Γ). If a is completely nondegenerate, then the set of singular points of VK∆(a) does not intersect KΓ and VK∆(a) is transversal to KΓ. The proof of this proposition is a fragment of the proof of Theorem 2.5.C. 2.5.E ((Corollary of 2.1.B and 2.1.C)). Let ∆ and a be as in 2.5.A. Then for any vector w ∈ C∆(∆) orthogonal to C∆(a)(∆(a)), a hypersurface VK∆(a) is invariant under transformations qhw,t : K∆ → K∆ and S(eπiw1 ,...,eπiwn) : K∆ → K∆ (the latter in the case of K = R is deﬁned only if w ∈ Zn). PATCHWORKING REAL ALGEBRAIC VARIETIES 29

3. Charts

3.1. Space R+∆. The aim of this Subsection is to distinguish in K∆ an important subspace which looks like ∆. More precisely, it is defined a stratified real semialge- braic variety R+∆, which is embedded in K∆ and homeomorphic, as a stratified space, to the polyhedron ∆ stratified by its faces. Briefly R+∆ can be described as the set of points with nonnegative real coordinates. If ∆ is a cone then R+∆ is defined as a subset of K∆ consisting of the points in which values of all monomials xω with ω ∈ ∆ ∩ Zn are real and nonnegative. It ′ is clear that for Γ ∈ G (∆) the set R+Γ coincides with R+∆ ∩ KΓ and for cones ∗ ∆1 ⊂ ∆2 a preimage of R+∆1 under in : K∆2 → K∆1 (see Section 2.3) is R+∆2. Now let ∆ be an arbitrary polyhedron. Embeddings, by which K∆ is glued form KC∆(Γ) with Γ ∈ G(∆), embed the sets R+C∆(Γ) in one another; a space obtained ′ by gluing from R+C∆(Γ) with Γ ∈ G(∆) is R+∆. It is clear that if Γ ∈ G then R+Γ= R+∆ ∩ KΓ. n n R+R is the open positive orthant {x ∈ RR | x1 > 0,...,xn > 0}. It can be n identified with the subgroup of quasi-homotheties of KR : one assigns qhl(x) to a n point x ∈ R+R . n n n n If A = {x ∈ R |x1 ≥ 0,...,xn ≥ 0} then KA = K (cf. Section 2.3) and n n R+A = A . If P n is the n-simplex with vertexes (0, 0,..., 0), (1, 0,..., 0), (0, 1, 0,..., 0), . . . , (0, 0,..., 1), then KP n is the n-simplex consisting of points of projective space with nonnegative real homogeneous coordinates. The set R+∆ is invariant under quasi-homotheties. Orbits of action in R+∆ of n the group of quasi-homotheties of R+R are sets R+CΓ(Γ) with Γ ∈ G(∆). Orbit dimΓ R+CΓ(Γ) is homeomorphic to R or, equivalently, to the interior of Γ. Closures R+Γ of R+CΓ(Γ) intersect one another in the same manner as the corresponding faces: R+Γ1 ∩R+Γ2 = R+(Γ1 ∩Γ2). From this and from the fact that R+Γ is locally conic (see [Loj64]) it follows that R+∆ is homeomorphic, as a stratified space, to ∆. However, there is an explicitly constructed homeomorphism. It is provided by the Atiyah moment map [Ati81] and in the case of bounded ∆ can be described in the following way. Choose a collection of points ω1,...,ωk with integer coordinates, whose convex hull is ∆. Then for Γ ∈ G(∆) and ω0 ∈ Γr∂Γ cone C∆(Γ) is hω1 −ω0,...,ωk −ω0i. ω ω n For y ∈ KC∆(Γ) denote by y a value of monomial x where ω ∈ C∆(Γ) ∩ Z at this point. Put k ωi−ω0 i=1 |y |ωi Rn M(y)= k ∈ . ωi−ω0 P i=1 |y | Obviously M(y) lies in ∆, does non depend on the choice of ω and for y ∈ P 0 KC∆(Γ1) ∩ KC∆(Γ2) does not depend on what face, Γ1 or Γ2, is used for the definition of M(y). Thus a map M : K∆ → ∆ is well defined. It is not difficult to R show that M|R+∆ : +∆ → ∆ is a stratified homeomorphism. Rn R Rn n 3.2. Charts of K∆. The space K can be presented as + × UK . In this Section an analogous representation of K∆ is described. R n +∆ is a fundamental domain for the natural action of UK in K∆, i.e. its intersection with each orbit of the action consists of one point. R n For a point x ∈ +CΓΓ where Γ ∈ G(∆), the stationary subgroup of action of UK consists of transformations S(eπiw1 ,...,eπiwn ), where vector (w1,...,wn) is orthogonal 30 OLEG VIRO

RP2R(P 1 P 1 )

Figure 26. to CΓ(Γ). In particular, if dim Γ = n then the stationary subgroup is trivial. If r n dim Γ = n − r then it is isomorphic to UK . Denote by UΓ a subgroup of UK πiw1 πiwn consisting of elements (e ,...,e ) with (w1,...,wn) ⊥ CΓ(Γ). R n Deﬁne a map ρ : +∆ × UK → K∆ by formula (x, y) 7→ Sy(x). It is surjection R n and we know the partition of +∆×UK into preimages of points. Since ρ is proper and K∆ is locally compact and Hausdorﬀ, it follows that K∆ is homeomorphic to R n the quotientspace of +∆×UK with respect to the partition into sets x×yUΓ with R n x ∈ +CΓ(Γ), y ∈ UK . Consider as an example the case of K = R and n = 2. Let a polyhedron ∆ lies 2 2 2 in the open positive quadrant. We place ∆ × UR in R identifying (x, y) ∈ ∆ × UR 2 2 2 with Sy(x) ∈ R . R+∆ × UR is homeomorphic ∆ × UR, so the surface R∆ can be obtained by an appropriate gluing (namely, by transformations taken from UΓ) 2 sides of four polygons consisting ∆ × UR. Figure 26 shows what gluings ought to be done in three special cases. 3.3. Charts of L-polynomials. Let a be a Laurent polynomial over K in n variables and ∆ be its Newton polyhedron. Let h be a homeomorphism ∆ → R+∆, mapping each face to the corresponding subspace, and such that for any Γ ∈ G(∆), n x ∈ Γ, y ∈ UK , z ∈ UΓ

R n h(x,y,z) = (pr +Γh(x, y), zprUK h(x, y)). − For h one can take, for example, (M|R+∆) . n A pair consisting of ∆ × UK and its subset υ which is the preimage of VK∆(a) under n h×id R n ρ ∆ × UK −−−−→ +∆ × UK −−−−→ K∆ is called a (nonreduced) K-chart of L-polynomial a. It is clear that the set υ is invariant under transformations id ×S with S ∈ U∆ n ′ and its intersection with Γ×UK , where Γ ∈ G (∆) is invariant under transformations id ×S with S ∈ UΓ. n As it follows from 2.5.A, if Γ is a face of ∆, and (∆ × UK , υ) is a nonreduced n n K-chart of L-polynomial a, then (Γ × UK , υ ∩ (Γ × UK )) is a nonreduced K-chart of L-polynomial aΓ. A nonreduced K-chart of Laurent polynomial a is unique up to homeomorphism n n ∆ × UK → ∆ × UK, satisfying the following two conditions: n (1) it map Γ × y with Γ ∈ G(∆) and y ∈ UK to itself and PATCHWORKING REAL ALGEBRAIC VARIETIES 31

n (2) its restriction to Γ × UK with g ∈ G(∆) commutes with transformations n n id ×S :Γ × UK → Γ × UK where S ∈ UΓ. In the case when a is a usual polynomial, it is convenient to place its K-chart n n n n into K . For this, consider a map A × UK → K : (x, y) 7→ Sy(x). Denote by n ∆K (a) the image of ∆(a) × UK under this map. Call by a (reduced) K-chart of a the image of a nonreduced K-chart of a under this map. The charts of peripherally nondegenerate real polynomial in two variables introduced in Section 1.3 are R- charts in the sense of this deﬁnition. 3.3.A. Let a be a Laurent polynomial over K in n variables, Γ a face of its New- R n ton polyhedron, ρ : +∆(a) × UK → K∆(a) a natural projection. If the truncation aΓ is completely nondegenerate then the set of singular points of hypersurface −1 R n R n −1 ρ VK∆(a)(a) of +∆(a) × UK does not intersect +Γ × UK, and ρ VK∆(a)(a) R n is transversal to +Γ × UK . Proof. Let ∆ be a resolution of polyhedron ∆(a). Then a commutative diagram ′ R n ′−1 ρ ( +∆ × UK , ρ (VK∆(a))) −−−−→ (K∆, VK∆(a))

(R+s×id) s

R n  −1 ρ  ( +∆(a) × U , ρ (VK∆(a)(a))) −−−−→ (K∆(a), VK∆(a)(a)) K y y appears. Here s is the natural regular map resolving singularities of K∆(a), ρ and ′ ρ are natural projections and R+s is a map R+∆ → R+∆(a) defined by s. The ′ preimage of KΓ under ρ is the union of KΣ with Σ ∈ G (∆) and C∆(Σ) ⊃ C∆(a)(Γ). By 2.5.D, the set of singular points of VK∆(a) does not intersect KΣ, and VK∆(a) is transversal to KΣ. ′ If Σ ∈ G (∆), C∆(Σ) ⊃ C∆(a)(Γ) and dim Σ = dim Γ, then R+s defines an ′ isomorphism R+CΣ(Σ) → R+CΓ(Γ), and if Σ ∈ G (∆), C∆(Σ) ⊃ C∆(a)(Γ) and dim Σ > dim Γ, then R+s defines a map R+CΣ(Σ) → R+CΓ(Γ) which is a factor- ization by the action of quasi-homotheties qhw,t with w ∈ CΣ(Σ), w ⊥ CΓ(Γ). By Γ 2.5.E, in the latter case variety VKΣ(a ) coinciding, by 2.5.A, with VK∆(a) ∩ KΣ −1 is invariant under the same quasi-homotheties. Hence VK∆(a) = s VK∆(a)(a) −1 ′−1 and hypersurface ρ VK∆(a), being the image of ρ VK∆(a) under R+ × id, ap- R n pears to be nonsingular along its intersection with +Γ × UK and transversal to R n +Γ × UK . 32 OLEG VIRO

4. Patchworking n 4.1. Patchworking L-polynomials. Let ∆, ∆1,...,∆s ⊂ R be (convex integer) s ∅ R polyhedra with ∆ = i=1 ∆i and Int ∆i ∩ Int ∆j = for i 6= j. Let ν : ∆ → be a nonnegative convex function satisfying to the following conditions: S (1) all the restrictions ν|∆i are linear; (2) if the restriction of ν to an open set is linear then this set is contained in one of ∆i; (3) ν(∆ ∩ Zn) ⊂ Z. Remark 4.1.A. Existence of such a function ν is a restriction on a collection ∆1,..., ∆s. For example, the collection of convex polygons shown in Figure 27 does not admit such a function.

Figure 27.

Let a1,...,as be Laurent polynomials over K in n variables with ∆(ai)=∆. Let ∆i∩∆j ∆i∩∆j ai = aj for any i, j. Then, obviously, there exists an unique L-polynomial ∆i ω a with ∆(a) = ∆ and a = ai for i =1,...,s. If a(x1,...,xn)= ω∈Zn aωx , we ω ν(ω) put b(x, t)= n a x t . This L-polynomial in n + 1 variables is considered ω∈Z ω P below also as a one-parameter family of L-polynomials in n variables. Therefore P let me introduce the corresponding notation: put bt(x1,...,xn)= b(x1,...,xn,t). L-polynomials bt are said to be obtained by patchworking L-polynomials a1,...,as by ν or, brieﬂy, bt is a patchwork of L-polynomials a1,...,as by ν.

4.2. Patchworking charts. Let a1,...,as be Laurent polynomials over K in n ∅ n variables with Int ∆(ai) ∩ Int ∆(aj ) = for i 6= j. A pair (∆ × UK, υ) is said to be obtained by patchworking K-charts of Laurent polynomials a1,...,as and it is s a patchwork of K-charts of L-polynomials a1,...,as if ∆ = i=1 ∆(ai) and one n can choose K-charts (∆(ai) × UK,υi) of Laurent polynomials a1,...,as such that s S υ = i=1 υi.

4.3. SThe Main Patchwork Theorem. Let ∆, ∆1,...,∆s, ν, a1,..., as, b and bt be as in Section 4.1 (bt is a patchwork of L-polynomials a1,..., as by ν).

4.3.A. If L-polynomials a1,...,as are completely nondegenerate then there exists t0 > 0 such that for any t ∈ (0,t0] a K-chart of L-polynomial bt is obtained by patchworking K-charts of L-polynomials a1,...,as. s ˜ Proof. Denote by G the union i=1 G(∆i). For Γ ∈ G denote by Γ the graph of ν|Γ. It is clear that ∆(b) is the convex hull of graph of ν, so Γ˜ ∈ G(∆(b)) and thus there S is an injection G → G(∆(b)) :Γ 7→ Γ.˜ Restrictions Γ˜ → Γ of the natural projection pr : Rn+1 → Rn are homeomorphisms, they are denoted by g. n+1 Let p : ∆(b) × UK → K∆(b) be the composition of the homeomorphism n+1 h×id R n+1 ∆(b) × UK −−−−→ +∆(b) × UK PATCHWORKING REAL ALGEBRAIC VARIETIES 33

R n+1 and the natural projection ρ : +∆(b) × UK → K∆(b) (cf. Section 3.3), so the n+1 −1 pair ∆(b) × UK , p VK∆(b)(b) is a K-chart of b. By 2.5.A, for i =1,...,s the pair ^ n+1 −1 ^ n ∆(ai) × UK ,p (VK∆(b)(b) ∩ ∆(ai) × UK ) ^ is a K-chart of L-polynomial b∆(ai). The pair ^ n −1 ^ n ∆(ai) × UK , p (VK∆(b)(b) ∩ ∆(ai) × UK )

^ n ^ n which is cut out by this pair on ∆(ai)×UK is transformed by g ×id : ∆(ai)×UK → n ^ ∆(ai) × UK to a K-chart of ai. Indeed, g : ∆(ai) → ∆(ai) deﬁnes an isomorphism ^ ∗ ^ ∆(ai) g : K∆(ai) → K∆(ai) and since b (x1,...,xn, 1) = ai(x1,...,xn), it follows ^ ∗ ∆(ai) that g : VK∆(ai)(ai)= V ^ (b ) and g deﬁnes a homeomorphism of the pair K∆(ai) ^ n −1 ^ n ∆(ai) × UK , p (VK∆(b)(b) ∩ ∆(ai) × UK ) to a K-chart of L-polynomial ai. Therefore the pair s s ^ n −1 ^ n ∆(ai) × UK , p (VK∆(b)(b) ∩ ∆(ai) × UK ) i=1 i=1 ! [ [ is a result of patchworking K-charts of a1,...,as. For t> 0 and Γ ∈ G′(∆) let us construct a ring homomorphism −1 n+1 n K[C∆(b)(∆(b) ∩ pr (Γ)) ∩ Z ] → K[C∆(Γ) ∩ Z ]

ω1 ωn ωn+1 ωn+1 ω1 ωn which maps a monomial x1 ...xn xn+1 to t x1 ...xn . This homomorphism corresponds to the embedding −1 KC∆(Γ) → KC∆(b)(∆(b) ∩ pr (Γ)) extending the embedding n n+1 KR → KR : (x1,...,xn) 7→ (x1,...,xn,t) . The embeddings constructed in this way agree to each other and deﬁne an embedding K∆ → K∆(b). Denote the latter embedding by it. It is clear that −1 VK∆(bt)= it VK∆(b)(b). −1 n+1 The sets ρ itK∆ are smooth hypersurfaces of ∆(b)×UK , comprising a smooth −1 1 isotopy. When t → 0, the hypersurface ρ itK∆ tends (in C -sense) to s ^ n ∆(ai) × UK . i=1 [ −1 By 3.3.A, ρ VK∆(b)(b) is transversal to each of R ^ n+1 +∆(ai) × UK −1 −1 and hence, the intersection ρ (itK∆) ∩ ρ (VK∆(b)(b)) for suﬃciently small t is mapped to s ^ n VK∆(b)(b) ∩ ∆(ai) × UK i=1 [ 34 OLEG VIRO by some homeomorphism s −1 ^ n ρ itK∆ → ∆(ai) × UK. i=1 [ Thus the pair −1 −1 −1 ρ itK∆, ρ itK∆ ∩ ρ VK∆(b)(b) is a result of patchworking K-charts of L-polynomials a1,...,as if t belongs to a −1 segment of the form (0,t0]. On the other hand, since VK∆(bt)= it VK∆˜ (b), −1 −1 −1 ρ itK∆ ∩ ρ VK∆(b)(b)= ρ itVK∆(bt) and, hence, the pair −1 −1 −1 ρ itK∆, ρ itK∆ ∩ ρ VK∆(b)(b) is homeomorphic to a K-chart of L-polynomial bt. PATCHWORKING REAL ALGEBRAIC VARIETIES 35

5. Perturbations smoothing a singularity of hypersurface The construction of the previous Section can be interpreted as a purposeful smoothing of an algebraic hypersurface with singularities, which results in replac- ing of neighborhoods of singular points by new fragments of hypersurface, having a prescribed topological structure (cf. Section 1.10). According to well known theorems of theory of singularities, all theorems on singularities of algebraic hypersurfaces are extended to singularities of signiﬁcantly wider class of hypersurfaces. In particular, the construction of perturbation based on patchworking is applicable in more general situation. For singularities of simplest types this construction together with some results of topology of algebraic curves allows to get a topological classiﬁcation of perturbations which smooth singularities completely. The aim if this Section is to adapt patchworking to needs of singularity theory.

5.1. Singularities of hypersurfaces. Let G ⊂ Kn be an open set, and let ϕ : G → K be an analytic function. For U ⊂ G denote by VU (ϕ) the set {x ∈ U | ϕ(x)= 0}. By singularity of a hypersurface VG(ϕ) at the point x0 ∈ VG(ϕ) we mean the class of germs of hypersurfaces which are diﬀeomorphic to the germ of VG(ϕ) at x0. In other words, hypersurfaces VG(ϕ) and VH (ψ) have the same singularity at points x0 and y0, if there exist neighborhoods M and N of x0 and y0 such that the pairs (M, VM (ϕ)), (N, VN (ψ)) are diﬀeomorphic. When considering a singularity of hypersurface at a point x0, to simplify the formulas we shall assume that x0 = 0. The multiplicity or the Milnor number of a hypersurface VG(ϕ) at 0 is the dimension

dimK K[[x1,...,xn]]/(∂f/∂x1,...,∂f/∂xn) of the quotient of the formal power series ring by the ideal generated by partial derivatives ∂f/∂x1,...,∂f/∂xn of the Taylor series expansion f of the function ϕ at 0. This number is an invariant of the singularity (see [AVGZ82]). If it is finite, then we say that the singularity is of finite multiplicity. If the singularity of VG(ϕ) at x0 is of finite multiplicity, then this singularity is n isolated, i.e. there exists a neighborhood U ⊂ K of x0, which does not contain singular points of VG(ϕ). If K = C then the converse is true: each isolated singularity of a hypersurface is of finite multiplicity. In the case of isolated singularity, n the boundary of a ball B ⊂ K , centered at x0 and small enough, intersects VG(ϕ) only at nonsingular points and only transversely, and the pair (B, VB (ϕ)) is homeomorphic to the cone over its boundary (∂B,V∂B (ϕ)) (see [Mil68], Theorem 2.10). In such a case the pair (∂B,V∂B (ϕ)) is called the link of singularity of VG(ϕ) at x0. The following Theorem shows that the class of singularities of finite multiplicity of analytic hypersurfaces coincides with the class of singularities of finite multiplicity of algebraic hypersurfaces. 5.1.A (Tougeron’s theorem). (see, for example, [AVGZ82], Section 6.3). If the singularity at x0 of a hypersurface VG(ϕ) has finite Milnor number µ, then there n exist a neighborhood U of x0 in K and a diffeomorphism h of this neighborhood n onto a neighborhood of x0 in K such that h(VU (ϕ)) = Vh(U)(f(µ+1)), where f(µ+1) is the Taylor polynomial of ϕ of degree µ + 1 . The notion of Newton polyhedron is extended over in a natural way to power ω series. The Newton polyhedron ∆(f) of the series f(x) = ω∈Zn aωx (where P 36 OLEG VIRO

ω ω1 ω2 ωn Rn x = x1 x2 ...xn ) is the convex hull of the set {ω ∈ | aω 6=0}. (Contrary to the case of a polynomial, the Newton polyhedron ∆(f) of a power series may have infinitely many faces.) However in the singularity theory the notion of Newton diagram occurred to be more important. The Newton diagram Γ(f) of a power series f is the union of the proper faces of the Newton polyhedron which face the origin, i.e. the union of the ′ + faces Γ ∈ G (∆(f)) for which cones D C∆(f)(Γ) intersect the open positive orthant n n Int A = {x ∈ R | x1 > 0,...,xn > 0}. It follows from the definition of the Milnor number that, if the singularity of VG(ϕ) at 0 is of finite multiplicity, the Newton diagram of the Taylor series of ϕ is compact, and its distance from each of the coordinate axes is at most 1. ω Rn For a power series f(x) = ω∈Zn fωx and a set Γ ⊂ the power series ω Γ n f x is called Γ-truncation of f and denoted by f (cf. Section 2.1). ω∈Γ∩Z ω P Let the Newton diagram of the Taylor series f of a function ϕ be compact. Then PΓ(f) n f is a polynomial. The pair (Γ(f) × UK,γ) is said to be a nonreduced chart Γ(f) n of germ of hypersurface VG(ϕ) at 0 if there exists a K-chart (∆(f × UK ,υ) of Γ(f) n f such that γ = υ ∩ (Γ(f) × UK ). It is clear that a nonreduced chart of germ of hypersurface is comprised of K-charts of f Γ, where Γ runs over the set of all faces of the Newton diagram. A power series f is said to be nondegenerate if its Newton diagram is compact and the distance between it and each of the coordinate axes is at most 1 and for any its face Γ polynomial f Γ is completely nondegenerate. In this case about the germ of VG(ϕ) at zero we say that it is placed nondegenerately. It is not difficult to prove that nondegenerately placed germ defines a singularity of finite multiplicity. It is convenient to place the charts of germs of hypersurfaces in Kn by a natural map n n n A ×UK → K : (x, y) 7→ Sy(x) (like K-charts of an L-polynomial, cf. Section 3.3). n Denote by ΣK (ϕ) the image of Γ(f)×UK under this map; the image of nonreduced chart of germ of hypersurface VG(ϕ) at zero under this map is called a (reduced) chart of germ of VG(ϕ) at the origin. It follows from Tougeron’s theorem that in mi this case adding a monomial of the form xi to ϕ with mi large enough does not change the singularity. Thus, without changing the singularity, one can make the Newton diagram meeting the coordinate axes. In the case when this takes place and the Taylor series of ϕ is nondegenerate there n exists a ball U ⊂ K centered at 0 such that the pair (U, VU (ϕ)) is homeomorphic to the cone over a chart of germ of VG(ϕ). This follows from Theorem 5.1.A and from results of Section 2.5. Thus if the Newton diagram meets all coordinate axes and the Taylor series of ϕ is nondegenerate, then the chart of germ of VG(ϕ) at zero is homeomorphic to the link of the singularity. 5.2. Evolving of a singularity. Now let the function ϕ : G → K be included as ϕ0 in a family of analytic functions ϕt : G → K with t ∈ [0,t0]. Suppose that this is an analytic family in the sense that the function G × [0,t0] → K : (x, t) 7→ ϕt(x) which is determined by it is real analytic. If the hypersurface VG(ϕ) has an isolated singularity at x0, and if there exists a neighborhood U of x0 such that the hypersurfaces VG(ϕt) with t ∈ [0,t0] have no singular points in U, then the family of functions ϕt with t ∈ [0,t0] is said to evolve the singularity of VG(ϕ) at x0. If the family ϕt with t ∈ [0,t0] evolves the singularity of the hypersurface VG(ϕ0) n at x0, then there exists a ball B ⊂ K centered at x0 such that PATCHWORKING REAL ALGEBRAIC VARIETIES 37

(1) for t ∈ [0,t0] the sphere ∂B intersects VG(ϕt) only in nonsingular points of the hypersurface and only transversely, (2) for t ∈ (0,t0] the ball B contains no singular point of the hypersurface VG(ϕt), (3) the pair (B, VB (ϕ0)) is homeomorphic to the cone over its boundary (∂B, V∂B (ϕ0)).

Then the family of pairs (B, VB (ϕt)) with t ∈ [0,t0] is called an evolving of the germ of VG(ϕ0) in x0. (Following the standard terminology of the singularity theory, it would be more correct to say not a on family of pairs, but rather a family of germs or even germs of a family; however, from the topological viewpoint, which is more natural in the context of the topology of real algebraic varieties, the distinction between a family of pairs satisfying 1 and 2 and the corresponding family of germs is of no importance, and so we shall ignore it.) Conditions 1 and 2 imply existence of a smooth isotopy ht : B → B with t ∈

(0,t0], such that ht0 = id and ht(VB (ϕt0 )) = VB (ϕt), so that the pairs (B, VB (ϕt)) with t ∈ (0,t0] are homeomorphic to each other. Given germs determining the same singularity, a evolving of one of them obviously corresponds to a diﬀeomorphic evolving of the other germ. Thus, one may speak not only of evolvings of germs, but also of evolvings of singularities of a hypersurface. The following three topological classiﬁcation questions on evolvings arise.

5.2.A. Up to homeomorphism, what manifolds can appear as VB (ϕt) in evolvings of a given singularity?

5.2.B. Up to homeomorphism, what pairs can appear as (B, VB (ϕt)) in evolvings of a given singularity?

′ ′ ′ ′ Smoothings (B, VB (ϕt)) with t ∈ [0,t0] and (B , VB (ϕt)) with t ∈ [0,t0] are ′ said to be topologically equivalent if there exists an isotopy ht : B → B with ′ ′ ′ t ∈ [0, min(t0,t0)], such that h0 is a diffeomorphism and VB (ϕt) = htVB (ϕt) for ′ t ∈ [0, min(t0,t0)]. 5.2.C. Up to topological equivalence, what are the evolvings of a given singularity? Obviously, 5.2.B is a refinement of 5.2.A. In turn, 5.2.C is more refined than 5.2.B, since in 5.2.C we are interested not only in the type of the pair obtained in result of the evolving, but also the manner in which the pair is attached to the link of the singularity. In the case K = R these questions have been answered in literature only for several simplest singularities. In the case K = C a evolving of a given singularity is unique from each of the three points of view, and there is an extensive literature (see, for example, [GZ77]) devoted to its topology (i.e., questions 5.2.A and 5.2.B). By the way, if we want to get questions for K = C which are truly analogous to questions 5.2.A — 5.2.C for K = R, then we have to replace evolvings by deforma- tions with nonsingular fibers and one-dimensional complex bases, and the variety VB(ϕt) and the pairs (B, VB (ϕt)) have to be considered along with the monodromy transformations. It is reasonable to suppose that there are interesting connections between questions 5.2.A — 5.2.C for a real singularity and their counter-parts for the complexification of the singularity. 38 OLEG VIRO

5.3. Charts of evolving. Let the Taylor series f of function ϕ : G → K be nondegenerate and its Newton diagram meets all the coordinate axes. Let a family of functions ϕt : G → K with t ∈ [0,t0] evolves the singularity of VG(ϕ) at 0. Let (B, VB (ϕt)) be the corresponding evolving of the germ of this hypersurface and ht : B → B with t ∈ (0,t0] be an isotopy with ht0 = id and ht(VB (ϕt0 )) = VB (ϕt)) existing by conditions 1 and 2 of the previous Section. Let (ΣK (ϕ),γ) be a chart of germ of hypersurface VG(ϕ) at zero and g : (ΣK (ϕ),γ) → (∂B,V∂B(ϕ)) be the natural homeomorphism of it to link of the singularity. n Denote by ΠK (ϕ) a part of K bounded by ΣK (ϕ). It can be presented as a cone over ΣK(ϕ) with vertex at zero. One can choose the isotopy ht : B → B, t ∈ (0,t0] such that its restriction to ′ ∂B can be extended to an isotopy ht : ∂B → ∂B with t ∈ [0,t0] (i.e., extended for t = 0). We shall call the pair (ΠK (ϕ), τ) a chart of evolving (B, VB(ϕt)), t ∈ [0,t0], if there exists a homeomorphism (ΠK (ϕ), τ) → (B, VB (t0)), whose restriction ′ h Σ (ϕ) → ∂B is the composition g 0 . One can see that K ΣK (ϕ) −−−−→ ∂B −−−−→ ∂B the boundary (∂ΠK (ϕ), ∂τ) of a chart of evolving is a chart (ΣK (ϕ), γ) of the germ of the hypersurface at zero, and a chart of evolving is a pair obtained by evolving which is glued to (ΣK (ϕ), γ) in natural way. Thus that the chart of an evolving describes the evolving up to topological equivalence. 5.4. Construction of evolvings by patchworking. Let the Taylor series f of function ϕ : G → K be nondegenerate and its Newton diagram Γ(f) meets all the coordinate axes. Let a1,...,as be completely nondegenerate polynomials over K in n variables ∅ ∆(ai)∩∆(aj ) ∆(ai)∩∆(aj ) with Int ∆(ai) ∩ Int ∆(aj ) = and ai = aj for i 6= j. Let s i=1 ∆(ai) be the polyhedron bounded by the coordinate axes and Newton dia- ∆(a )∩∆(f) s gram Γ(f). Let a i = f ∆(ai)∩∆(f) for i =1,...,s. Let ν : ∆(a ) → R S i i=1 i be a nonnegative convex function which is equal to zero on Γ(f) and satisfies con- S ditions 1, 2, 3 of Section 4.1 with polyhedra ∆(a1), . . . , ∆(as). Then polynomials a1,...,as can be ”glued to ϕ by ν” in the following way generalizing patchworking L-polynomials of Section 4.1. ∆(ai) Denote by a the polynomial defined by conditions a = ai for i = 1,...,s s Si=1 ∆(ai) ω and a = a. If a(x)= ω∈Zn aωx then we put ω ν(ω) Γ(f) ϕt(x)= ϕ(Px) + ( aωx t ) − a x. ω Zn X∈ 5.4.A. Under the conditions above there exists t0 > 0 such that the family of functions ϕt : G → K with t ∈ [0,t0] evolves the singularity of VG(ϕ) at zero. The chart of this evolving is patchworked from K-charts of a1,...,as. In the case when ϕ is a polynomial, Theorem 5.4.A is a slight modification of a special case of Theorem 4.3.A. Proof of 4.3.A is easy to transform to the proof of this version of 5.4.A. The general case can be reduced to it by Tougeron Theorem, or one can prove it directly, following to scheme of proof of Theorem 4.3.A. We shall call the evolvings obtained by the scheme described in this Section patchwork evolvings. PATCHWORKING REAL ALGEBRAIC VARIETIES 39

6. Approximation of hypersurfaces of KRn 6.1. Sufficient truncations. Let M be a smooth submanifold of a smooth manifold X. Remind that by a tubular neighborhood of M in X one calls a submanifold N of X with M ⊂ Int N equipped with a tubular fibration, which is a smooth retraction p : N → M such that for any point x ∈ M the preimage p−1(x) is a smooth submanifold of X diffeomorphic to Ddim X−dim M . If X is equipped with a metric and each fiber of the tubular fibration p : N → M is contained in a ball of radius ε centered in the point of intersection of the fiber with M, then N is called a tubular ε-neighborhood of M in X. We need tubular neighborhoods mainly for formalizing a notion of approximation of a submanifold by a submanifold. A manifold presented as the image of a smooth section of the tubular fibration of a tubular ε-neighborhood of M can be considered as sufficiently close to M: it is naturally isotopic to M by an isotopy moving each point at most by ε. Rn n We shall consider the space × UK as a flat Riemannian manifold with metric defined by the standard Euclidian metric of Rn in the case of K = R and by the standard Euclidian metric of Rn and the standard flat metric of the torus n 1 n UC = (S ) in the case of K = C. An ε-sufficiency of truncations of Laurent polynomial defined below and the whole theory related with this notion presuppose that it has been chosen a class of Rn n tubular neighborhoods of smooth submanifolds of × UK invariant under trans- n ′ lations Tω × idUK and that for any two tubular neighborhoods N and N of the same M, which belong to this class, restrictions of tubular fibrations p : N → M and p′ : N ′ → M to N ∩ N ′ coincide. One of such classes is the collection of all normal tubular neighborhoods, i.e. tubular neighborhoods with fibers consisting of segments of geodesics which start from the same point of the submanifold in directions orthogonal to the submanifold. Another class, to which we shall turn in Sections 6.7 and 6.8, is the class of tubular neighborhoods whose Rn−1 n−1 Rn n fibers lie in fibers × t × UK × s of × UK and consist of segments of geodesics which are orthogonal to intersections of the corresponding manifolds Rn−1 n−1 with these × t × UK × s. The intersection of such a tubular neighbor- Rn−1 n−1 hood of M with the fiber × t × UK × s is a normal tubular neighborhood Rn−1 n−1 Rn−1 n−1 of M ∩ ( × t × UK × s) in × t × UK × s. Of course, only manifolds Rn−1 n−1 transversal to × t × UK × s have tubular neighborhoods of this type. Introduce a norm in vector space of Laurent polynomials over K on n variables:

ω n || aωx || = max{|aω| | ω ∈ Z }. ω Zn X∈ Let Γ be a subset of Rn and ε a positive number. Let a be a Laurent polynomial over K in n variables and U a subset of KRn. We shall say that in U the truncation aΓ is ε-suﬃcient for a (with respect to the chosen class of tubular neighborhoods), if for any Laurent polynomial b over K satisfying the conditions ∆(b) ⊂ ∆(a), bΓ = aΓ and ||b − bΓ|| ≤ ||a − aΓ|| (in particular, for b = a and b = aΓ) the following condition takes place:

(1) U ∩ SVKRn (b)= ∅, (2) the set la(U ∩VKRn (b)) lies in a tubular ε-neighborhood N (from the chosen Γ Γ class) of la(VKRn (a ) r SVKRn (a )) and 40 OLEG VIRO

(3) la(U ∩ VKRn (b)) can be extended to the image of a smooth section of the Γ Γ tubular fibration N → la(VKRn (a ) r SVKRn (a )). The ε-sufficiency of Γ-truncation of Laurent polynomial a in U means, roughly Γ speaking, that monomials which are not in a have a small influence on VKRn (a)∩U.

Γ 6.1.A. If a is ε-sufficient for a in open sets Ui with i ∈ J , then it is ε-sufficient for a in i∈J Ui too. StandardS arguments based on Implicit Function Theorem give the following The- orem. 6.1.B. If a set U ⊂ KRn is compact and contains no singular points of a hypersurface VKRn (a), then for any tubular neighborhood N of VKRn (a) r SVKRn (a) and any polyhedron ∆ ⊃ ∆(a) there exists δ > 0 such that for any Laurent polynomial b with ∆(b) ⊂ ∆ and ||b − a|| <δ the hypersurface VKRn (b) has no singularities in U, intersection U ∩ VKRn (b) is contained in N and can be extended to the image of a smooth section of a tubular fibration N → VKRn (a) r SVKRn (a). From this the following proposition follows easily. 6.1.C. If U ∈ KRn is compact and aΓ is ε-sufficient truncation of a in U, then for any polyhedron ∆ ⊃ ∆(a) there exists δ > 0 such that for any Laurent polynomial b with ∆(b) ⊂ ∆, bΓ = aΓ and ||b − a|| <δ the truncation bΓ is ε-sufficient in U. In the case of Γ = ∆(a) proposition 6.1.C turns to the following proposition.

n 6.1.D. If a set U ⊂ KR is compact and contains no singular points of VKRn (a) and la(VKRn (a)) has a tubular neighborhood of the chosen type, then for any ε > 0 and any polyhedron ∆ ⊃ ∆(a) there exists δ > 0 such that for any Laurent polynomial b with ∆(b) ⊂ ∆, ||b − a|| < δ and b∆(a) = a the truncation b∆(a) is ε-sufficient in U. The following proposition describes behavior of the ε-sufficiency under quasi- homotheties. 6.1.E. Let a be a Laurent polynomial over K in n variables. Let U ⊂ KRn, Γ ⊂ Rn, w ∈ Rn. Let ε and t be positive numbers. Then ε-sufficiency of Γ-truncation aΓ of a in qhw,t(U) is equivalent to ε-sufficiency of Γ-truncation of a ◦ qhw,t in U. The proof follows from comparison of the definition of ε-sufficiency and the following two facts. First, it is obvious that −1 qhw,t(U) ∩ VKRn (b)= qhw,t(U ∩ qhw,t(VKRn (b))) = qhw,t(U ∩ VKRn (b ◦ qhw,t)),

n Rn n and second, the transformation T(ln t)w × idUK of × UK corresponding, by 2.1.A, to qhw,t preserves the chosen class of tubular εneighborhoods .

6.2. Domains of ε-suﬃciency of face-truncation. For A ⊂ Rn and B ⊂ KRn denote by qhA(B) the union ω∈A qhω(B). n n For A ⊂ R and ρ> 0 denote by Nρ(A) the set {x ∈ R |dist(x, A) <ρ}. For A, B ⊂ Rn and λ ∈ RSthe sets {x + y | x ∈ A, y ∈ B} and {λx | x ∈ A} are denoted, as usually, by A + B and λA. Let a be a Laurent polynomial in n variables, ε a positive number and Γ a face of the Newton polyhedron ∆ = ∆(a). PATCHWORKING REAL ALGEBRAIC VARIETIES 41

6.2.A. If in open set U ⊂ KRn the truncation aΓ is ε-sufficient for a, then it is 5 ε-sufficient for a in qh − (U). Cl DC∆ (Γ) − Proof. Let ω ∈ Cl DC∆(Γ) and ωw = δ for w ∈ Γ. By 6.1.E, ε-sufficiency of Γ Γ truncation a for a in qhω(U) is equivalent to ε-sufficiency of truncation (a ◦ qhω) Γ for (a ◦ qhω) in U or, equivalently, to ε-sufficiency of Γ-truncation of Laurent −δ polynomial b = e a ◦ qhω in U. Since −δ −δ w ωw Γ ωw−δ w e a ◦ qhω(x)= e awx e = a (x)+ e awx w w r X∈∆ ∈X∆ Γ r − and ωw − δ ≤ 0 when w ∈ ∆ Γ and ω ∈ Cl DC∆(Γ), it follows that b satisfies the conditions ∆(b) = ∆, bΓ = aΓ and ||b − bΓ|| ≤ ||a − aΓ||. Therefore the truncation bΓ is ε-sufficient for b in U and, hence, the truncation aΓ is ε-sufficient for a in qhω(U). From this, by 6.1.A, the proposition follows.

Γ Γ 6.2.B. If the truncation a is completely nondegenerate and laVKRn (a ) has a tubular neighborhood of the chosen type, then for any compact sets C ⊂ KRn and − Γ Ω ⊂ DC∆(Γ) there exists δ such that in qhδΩ(C) the truncation a is ε-sufficient for a. − Proof. For ω ∈ DC∆(Γ) denote by ωΓ a value taken by the scalar product ωw for w ∈ Γ. Since −ωΓ Γ ωw−ωΓ w t a ◦ qhω,t(x)= a (x)+ t awx w r ∈X∆ Γ − r for ω ∈ DC∆(Γ) (cf. the previous proof) and ωw − ωΓ < 0 when w ∈ ∆ Γ − −ωΓ and ω ∈ DC∆ (Γ) it follows that the Laurent polynomial bω,t = t a ◦ qhω,t with − Γ ω ∈ DC∆ (Γ) turns to a as Γ → +∞. It is clear that this convergence is uniform − with respect to ω on a compact set Ω ⊂ DC∆ (Γ). By 6.1.D it follows from this that for a compact set U ⊂ KRn there exists η such that for any ω ∈ Ω and t ≥ η the Γ truncation bω,t of bω,t is ε-sufficient in U for bω,t. By 6.1.E, the latter is equivalent Γ to ε-sufficiency of truncation a for a in qhω,t(U). Thus if U is the closure of a bounded neighborhood W of a set C then there exists η such that for ω ∈ Ω and t ≥ η the truncation aΓ is ε-sufficient for a in Γ qhω,t(U). Therefore a is the same in a smaller set qhω,t(W ) and, hence, (by 6.1.A) in the union t≥η,ω∈Ω qhω,t(W ) and, hence, in a smaller set t=η,ω∈Ω(C). Putting δ = ln η we obtain the required result. S S 6.2.C. Let Γ is a face of another face Σ of the polyhedron ∆. Let Ω is a compact − Γ Σ subset of the cone DC∆ (Σ). If Γ-truncation a is ε-sufficient for a in a compact Γ set C, then there exists a number δ such that a is ε-sufficient for a in qhδΩ(C). This proposition is proved similarly to 6.2.B, but with the following difference: the reference to Theorem 6.1.D is replaced by a reference to Theorem 6.1.C. 6.2.D. Let C ⊂ KRn be a compact set and let Γ be a face of ∆ such that for any face Σ of ∆ with dim Σ = dim ∆ − 1 having a face Γ the truncation aΓ is ε-sufficient for aΣ in C. Then there exists a real number δ such that the truncation Γ a is ε-sufficient for a in qh − rN − (Int C). Cl DC∆ (Γ) δDC∆ (∆)

5Here (as above) Cl denotes the closure. 42 OLEG VIRO

Proof. By 6.2.C, for any face Σ of ∆ with dim Σ = dim ∆ − 1 and Γ ⊂ ∂Σ there − Γ exists a vector ωΣ ∈ DC∆(Σ) such that the truncation a is ε-sufficient for a in − qhωΣ (C), and, hence, by 6.2.A, in qh (Int C). Choose such ωΣ for each ωΣ+Cl DC∆ (Γ) − Σ with dim Σ = dim ∆ − 1 and Γ ⊂ ∂Σ. Obviously, the sets ωΣ + Cl DC∆(Γ) − cover the whole closure of the cone DC∆(Γ) besides some neighborhood of its − top, i.e. the cone DC∆(∆); in other words, there exists a number δ such that − − r N − Γ Σ(ωΣ + Cl DC∆(Γ)) ⊃ Cl DC∆(Γ) δDC∆ (∆). Hence, a is ε-sufficient for a − − − in qhCl DC (Γ)rN DC (∆)(Int C) ⊂ Σ qhω +Cl DC (Γ)(Int C). S ∆ δ ∆ Σ ∆ 6.3. The main Theorem on logarithmicS asymptotes of hypersurface. Let ∆ ⊂ Rn be a convex closed polyhedron and ϕ : G(∆) → R be a positive function. Then for Γ ∈ G(∆) denote by D∆,ϕ(Γ) the set N − r N − ϕ(Γ)(DC∆ (Γ)) ϕ(Σ)(DC∆ (Σ)). Σ∈G(∆)[, Γ∈G(Σ) n It is clear that the sets D∆,ϕ(Γ) with Γ ∈ G(∆) cover R . Among these sets only sets corresponding to faces of the same dimension can intersect each other. In some cases (for example, if ϕΓ grows fast enough when dimΓ grows) they do not n intersect and then {D∆,ϕ(Γ)}Γ∈G(∆) is a partition of R . Let a be a Laurent polynomial over K in n variables and ε be a positive number. A function ϕ : G(∆(a)) → Rn is said to be describing domains of ε-sufficiency for a (with respect to the chosen class of tubular neighborhoods) if for any proper face Γ ∈ G(∆(a)), for which truncation aΓ is completely non-degenerate and the Γ hypersurface la(VKRn (a )) has a tubular neighborhood of the chosen class, the Γ −1 truncation a is ε-sufficient for a in some neighborhood of l (DC∆(a),ϕ(Γ)). 6.3.A. For any Laurent polynomial a over K in n variables and ε> 0 there exists a function G(∆(a)) → R describing domains of ε-sufficiency for a with respect to the chosen class of tubular neighborhoods. In particular, if a is peripherally nondegenerate Laurent polynomial over K in n variables and dim ∆(a) = n then for any ε > 0 there exists a compact set C ⊂ KRn such that KRn r C is covered by regions in which truncations of a∂∆(a) are ε-sufficient for a with respect to class of normal tubular neighborhoods. In other words, under these conditions behavior of VKRn (a) outside C is defined by monomials of a∂∆(a). 6.4. Proof of Theorem 6.3.A. Theorem 6.3.A is proved by induction on dimension of polyhedron ∆(a). If dim ∆(a) = 0 then a is monomial and VKRn (a)= ∅. Thus for any ε> 0 any function ϕ : G(∆(a)) → R describes domains of ε-sufficiency for a. Induction step follows obviously from the following Theorem. 6.4.A. Let a be a Laurent polynomial over K in n variables, ∆ be its Newton polyhedron, ε a positive number. If for a function ϕ : G(∆) r {∆} → R and any proper face Γ of ∆ the restriction ϕ|G(Γ) describes domains of ε-sufficiency for aΓ, then ϕ can be extended to a function ϕ¯ : G(∆) → R describing regions of ε-sufficiency for a. Proof. It is sufficient to prove that for any face Γ ∈ G(∆) r {∆}, for which the Γ Γ truncation a is completely nondegenerate and hypersurface VKRn (a ) has a tubular PATCHWORKING REAL ALGEBRAIC VARIETIES 43 neighborhood of the chosen class, there exists an extension ϕΓ of ϕ such that Γ −1 truncation a is ε-sufficient for a in a neighborhood of l (D∆,ϕΓ (Γ)), i.e. to prove Γ that for any face Γ 6= ∆ there exists a number ϕΓ(∆) such that the truncation a is ε-sufficient for a in some neighborhood of −1 N − r N − N − l ( ϕ(Γ)(DC∆(Γ)) [ ϕΓ(∆)(DC∆(∆)) ϕ(Σ)(DC∆ (Σ))]. r Σ∈G(∆) {[∆}, Γ∈G(Σ) Indeed, putting

ϕ¯(∆) = max ϕΓ(∆) Γ∈G(∆)r{∆} we obtain a required extension of ϕ. First, consider the case of a face Γ with dimΓ = dim∆ − 1. Apply proposition −1 N − 6.2.B to C = l (Cl ϕ(Γ)+1(0) and any one-point set Ω ⊂ DC∆(Γ). It implies that Γ −1 N − a is ε-sufficient for a in qhω(C)= l (Cl ϕ(Γ)+1(ω)) for some ω ∈ DC∆ (Γ). Now −1 Γ apply proposition 6.2.A to U = l (Nϕ(Γ)+1(ω)). It gives that a is ε-sufficient −1 −1 − for a in qh − (l (Nϕ(Γ)+1(ω)) = l (Nϕ(Γ)+1(ω + DC (Γ))) and, hence, in DC∆ (Γ) ∆ −1 N − r N − the smaller set l ( ϕ(Γ)+1(DC∆(Γ))) |ω|(DC∆ (∆)). It is remained to put ϕΓ(∆) = |ω| + 1. Now consider the case of face Γ with dimΓ < dim ∆ − 1. Denote by E the set N − r N − ϕ(Γ)(DC∆ (Γ)) ϕ(Σ)(DC∆ (Σ)). r Σ∈G(∆) {[∆}, Γ∈G(Σ) It is clear that there exists a ball B ⊂ Rn with center at 0 such that E = (E ∩ B)+ − Cl DC∆ (Γ). Denote the radius of this ball by β. If Σ ∈ G(∆) is a face of dimension dim ∆−1 with ∂Σ ⊃ Γ then, by the hypothesis, the truncation aΓ is ε-sufficient for aΣ in some neighborhood of −1 N − r N − l ( ϕ(Γ)(DCΣ (Γ)) ϕ(Θ)(DCΣ (Θ)) Θ∈G(Σ)[, Γ∈G(Θ) and, hence, in neighborhood of a smaller set −1 N − r N − l ( ϕ(Γ)(DC∆ (Γ)) ϕ(Θ)(DC∆ (Θ)). Θ∈G(Σ)[, Γ∈G(Θ) Therefore for any face Σ with dimΣ = dim∆ − 1 and Γ ⊂ ∂Σ the truncation aΓ is ε-sufficient for aΣ in some neighborhood of l−1(E). Denote by C a compact neighborhood of l−1(E ∩ B) contained in this neighborhood. Applying proposition 6.2.A, one obtains that aΓ is ε-sufficient for a in the set −1 − − qh − rN − (Int C)= l (Int l(C) + Cl DC (Γ) r NδDC (∆))). Cl DC∆ (Γ) δDC∆ (∆) ∆ ∆

It is remained to put ϕΓ(∆) = δ + β 6.5. Modification of Theorem 6.3.A. Below in Section 6.8 it will be more convenient to use not Theorem 6.3.A but the following its modification, whose formulation is more cumbrous, and whose proof is obtained by an obvious modification of deduction of 6.3.A from 6.4.A. 6.5.A. For any Laurent polynomial a over K in n variables and any ε > 0 and c > 1 there exists a function ϕ : G(∆(a)) → R such that for any proper face Γ Γ Γ ∈ G(∆(a)), for which a is completely nondegenerate and la(VKRn (a )) has a 44 OLEG VIRO tubular neighborhood from the chosen class, the truncation aΓ is ε-sufficient for a in some neighborhood of −1 N − r N − l ( cϕ(Γ)(DC∆ (Γ)) ϕ(Σ)(DC∆(Σ)). Σ∈G(∆)[, Γ∈G(Σ)

6.6. Charts of L-polynomials. Let a be a peripherally nondegenerate Laurent polynomial over K in n variables, ∆ be its Newton polyhedron. Let V be a vector subspace of Rn corresponding to the smallest affine subspace containing ∆ (i.e. V = C∆(∆)). Let ϕ : G(∆) → R be the function, existing by 6.3.A, describing for some ε regions of ε-sufficiency for a with respect to class of normal tubular neighborhoods. n n The pair (∆ × UK , υ) consisting of the product ∆ × UK and its subset υ is a K-chart of a Laurent polynomial a if: n n (1) there exists a homeomorphism h : (Cl D∆,ϕ(∆) ∩ V ) × UK → ∆ × UK such n that h((Cl D∆,ϕ(∆) ∩ V ) × y) = ∆ × y for y ∈ UK, n υ = h(laVKRn (a) ∩ (Cl D∆,ϕ(∆) ∩ V ) × UK n and for each face Γ of ∆ the set h((Cl D∆,ϕ(∆) ∩ D∆,ϕ(Γ) ∩ V ) × UK ) lies n in the product of the star Γ∈G(Σ)Σ∈G(∆)r{∆} Σ ofΓ to UK ; (2) for any vector ω ∈ Rn, which is orthogonal to V and, in the case of K = R, S n n is integer, the set υ is invariant under transformation ∆ × UK → ∆ × UK πiω1 πiωn defined by formula (x, (y1,..., yn)) 7→ (x, (e y1, ..., e yn)); n n (3) for each face Γ of ∆ the pair (Γ×UK,υ ∩(Γ×UK)) is a K-chart of Laurent polynomial aΓ. The definition of the chart of a Laurent polynomial, which, as I believe, is clearer than the description given here, but based on the notion of toric completion of KRn, is given above in Section 3.3. I restrict myself to the following commentary of conditions 1 – 3. n The set (Cl D∆,ϕ(∆) ∩ V ) × UK contains, by 6.3.A, a deformation retract of laVKRn (a). Thus, condition 1 means that υ is homeomorphic to a deformation n retract of VKRn (a). The position of υ in ∆ × UK contains, by 1 and 3, a complete topological information about behavior of this hypersurface outside some compact set. The meaning of 2 is in that υ has the same symmetries as, according to 2.1.C, VKRn (a) has. n 6.7. Structure of VKRn (bt) with small t. Denote by it the embedding KR → n+1 KR defined by it(x1,...,xn) = (x1,...,xn, t). Obviously, −1 VKRn (bt)= it VKRn+1 (b).

This allows to take advantage of results of the previous Section for study of VKRn (bt) as t → 0. For suﬃciently small t the image of embedding it is covered by regions of ε-suﬃciency of truncation bΓ˜ , where Γ˜ runs over the set of faces of graph of ν, and therefore the hypersurface VKRn (bt) turns to be composed of pieces obtained from VKRn (ai) by appropriate quasi-homotheties. I preface the formulation describing in detail the behavior of VKRn (bt) with several notations. Denote the Newton polyhedron ∆(b) of Laurent polynomial b by ∆.˜ It is clear ˜ s that ∆ is the convex hull of the graph of ν. Denote by G the union i=1 G(∆i). For S PATCHWORKING REAL ALGEBRAIC VARIETIES 45

Γ ∈ G denote by Γ˜ the graph ν|Γ. It is clear that Γ˜ ∈ G(∆)˜ and hence an injection Γ → Γ:˜ G → G(∆)˜ is deﬁned. n n+1 For t > 0 denote by jt the embedding R → R deﬁned by the formula jt(x1,...,xn) = (x1,...,xn, ln t). Let ψ : G → R be a positive function, t be a number from interval (0, 1). For Γ ∈ G denote by Et,ψ(Γ) the following subset of Rn: N j−1(DC−(Γ))˜ r N j−1(DC−(Σ))˜ . ψ(Γ) t ∆˜ ϕ(Σ) t ∆˜ Σ∈G,[Γ∈G(Σ)

6.7.A. If Laurent polynomials a1,...,as are completely non-degenerate then for any ε> 0 there exist t0 ∈ (0, 1) and function ψ : G → R such that for any t ∈ (0,t0] Γ and any face Γ ∈ G truncation bt is ε-suﬃcient for bt with respect to the class of −1 normal tubular neighborhoods in some neighborhood of l (Et,ψ(Γ)).

Γ Denote the gradient of restriction of ν on Γ ∈ G by ∇(Γ). The truncation bt , Γ ∆i obviously, equals a ◦ qh∇(Γ),t. In particular, bt = ai ◦ qh∇(∆i),t and, hence,

∆i n −1 n VKR (bt )= qh∇(∆i),t (VKR (ai)).

Γ In the domain, where bt is ε-suﬃcient for bt, the hypersurfaces laVKRn (bt) ∆i and laVKRn (bt ) with ∆i ⊃ Γ lie in the same normal tubular ε-neighborhood Γ of laVKRn (bt ) and, hence, are isotopic by an isotopy moving points at most on 2ε. n Thus, according to 6.7.A, for t ≤ t0 to the space KR is covered by regions in n −1 n which VKR (bt) is approximated by qh∇(∆i),t (VKR (ai)).

2 6.8. Proof of Theorem 6.7.A. Put c = max{ 1+ ∇(∆i) ,i =1,...,s}. Apply Theorem 6.5.A to the Laurent polynomial b and numbers ε and c, considering as Rnp+1 n+1 the class of chosen tubular neighborhoods in × UK tubular neighborhoods, Rn n Rn+1 n+1 whose fibers lie in the fibers ×t×UK ×s of ×UK and consist of segments of Rn n geodesics which are orthogonal to intersections of submanifold with ×t×UK ×s. Rn+1 n+1 (Intersection of such a tubular neighborhood of M ⊂ × UK with the fiber Rn n Rn n × t × UK × s is a normal tubular neighborhood of M ∩ ( × t × UK × s) in Rn n ˜ R × t × UK × s.) Applying Theorem 6.5.A one obtains a function ϕ : G(∆) → . Denote by ψ the function G → R which is the composition of embedding Γ 7→ Γ:˜ 1 G → G(∆)˜ (see Section 6.7) and the function ϕ : G(∆)˜ → R. This function has c −ϕ(∆)˜ the required property. Indeed, as it is easy to see, for 0

j−1(N (DC−(Γ))˜ r N (DC−(Σ)))˜ , t cϕ(Γ)˜ ∆˜ ϕ(Σ)˜ ∆˜ ˜ ˜ ˜ ˜ Σ∈G(∆)[, Γ∈G(Σ) and thus from ε-suﬃciency of bΓ˜ for b in some neighborhood of

l−1(N (DC−(Γ))˜ r N (DC−(Σ)))˜ , cϕ(Γ)˜ ∆˜ ϕ(Σ)˜ ∆˜ ˜ ˜ ˜ ˜ Σ∈G(∆)[, Γ∈G(Σ) Rn+1 n+1 with respect to the chosen class of tubular neighborhoods in ×UK if follows −ϕ(∆)˜ Γ that for 0

6.9. An alternative proof of Theorem 4.3.A. Let V be a vector subspace of Rn corresponding to the minimal aﬃne subspace containing ∆. It is divided for each t ∈ (0, 1) onto the sets V ∩ j−1(DC−(Γ))˜ with Γ ∈ G. Let us construct cells Γ t ∆˜ t in V which are dual to the sets of this partition (barycentric stars). For this mark a point in each V ∩ j−1(DC−(Γ)):˜ t ∆˜ b ∈ V ∩ j−1(DC−(Γ))˜ . t,Γ t ∆˜ Then for Γ with dimΓ =0 put Γt = bt,Γ and construct the others Γt inductively on dimension dim Γ: if Γt for Γ with dim Γ < r have been constructed then for Γ with dim Γ = r the cell Γt is the (open) cone on Σ∈G(Γ)r{Γ} Σt with the vertex bt,Γ. (This is the usual construction of dual partition turning in the case of triangulation S to partition onto barycentric stars of simplices.) ′ R By Theorem 6.7.A there exist t0 ∈ (0, 1) and function ψ : G → such that for ′ Γ any t ∈ (0,t0] and any face Γ ∈ G the truncation bt is ε-suﬃcient for bt in some −1 neighborhood of l (Et,ψ(Γ)). Since cells Γt grow unboundedly when t runs to zero ′ (if dim Γ 6= 0) it follows that there exists t0 ∈ (0,t0] such that for t ∈ (0,t0] for each face Γ ∈ G the set N j−1(DC−(Γ)), and together with it the set E (Γ), ψ(Γ) t ∆˜ t,ψ lie in the star of the cell Γt, i.e. in Γ∈G(Σ) Σt. Let us show that for such t0 the conclusion of Theorem 4.3.A takes place. S n Indeed, it follows from 6.7.A that there exists a homeomorphism h :Γt × UK → n n n Γ × UK with h(Γt × y)=Γ × y for y ∈ UK such that (Γ × UK , h(la(VKRn (bt)) ∩ n Γ Γt × UK )) is K-chart of Laurent polynomial a . Therefore the pair n n (∪Γ∈G Γt × UK , laVKRn (bt) ∩ (∪Γ∈G Γt × UK )) is obtained in result of patchworking K-charts of Laurent polynomials a1, ..., as. The function ϕ : G(∆) → R, existing by Theorem 6.3.A applied to bt, can be chosen, as it follows from 6.4.A, in such a way that it should majorate any given in advance function G(∆) → R. Choose ϕ in such a way that Dϕ,∆(∆) ⊃ ∪Γ∈GΓt and Dϕ,∆(Σ) ∩ ∂Dϕ,∆(∆) ⊃Et,ψ(Σ) ∩ ∂Dϕ,∆(∆) for Σ ∈ G(∆) r {∆}. As it follows from 6.7.A, there exists a homeomorphism

n n (1) ( Γt × UK, laVKRn (bt) ∩ ( Γt × UK )) → Γ∈G Γ∈G [ [ n n (Dϕ,∆(∆) × UK , laVKRn (bt) ∩ (Dϕ,∆(∆) × UK )) n r turning Et,ψ(Σ) ∩ ∂( Γ∈G Γt × UK ) to Et,ψ(Σ) ∩ ∂Dϕ,∆(∆) for Σ ∈ G(∆) {∆}. Therefore K-chart of Laurent polynomial bt is obtained by patchworking K-charts S of Laurent polynomials a1,...,as. PATCHWORKING REAL ALGEBRAIC VARIETIES 47

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Mathematical Department, Uppsala University, S-751 06 Uppsala, Sweden; POMI, Fontanka 27, St. Petersburg, 191011, Russia E-mail address: oleg.viro@@gmail.com